Explanation:
Okay, so I need to find the perimeter of this set S. Let me try to understand what S is. It says S is the union of two things: first, all the points inside a regular nonagon with side length 2 units, and second, all the points less than 1 unit away from a point on the perimeter of the nonagon. So, essentially, S is the original nonagon plus a sort of "buffer" zone around it, right? Like, if you imagine taking the nonagon and then expanding it outward by 1 unit in all directions, but only up to the points that are within 1 unit of the perimeter. Wait, but actually, since it's the union of the interior of the nonagon and all points within 1 unit of the perimeter, that would be like taking the original nonagon and then adding a 1-unit wide strip around it. But actually, when you take all points within 1 unit of the perimeter, that's like taking the Minkowski sum of the perimeter with a disk of radius 1. But since the original nonagon is already included, maybe S is the original nonagon expanded by 1 unit? Hmm, but maybe not exactly, because when you take all points within 1 unit of the perimeter, that includes the original nonagon? Wait, no. If you take all points within 1 unit of the perimeter, that would include the original perimeter and the area inside the nonagon? Wait, no. Wait, if you take all points less than 1 unit away from the perimeter, that would include points inside the nonagon? Let me think.

Suppose you have a point inside the nonagon. Is it less than 1 unit away from the perimeter? Well, if the nonagon has a certain radius, then points near the center would be more than 1 unit away from the perimeter. So maybe S is the union of the interior of the nonagon and the area within 1 unit of the perimeter. So that would be like the original nonagon expanded outward by 1 unit, but also including the area inside the original nonagon? Wait, but if you take the union of the interior and the points within 1 unit of the perimeter, then the interior is already included in the points within 1 unit of the perimeter? Wait, no. If you take all points within 1 unit of the perimeter, that would include points inside the nonagon that are within 1 unit of the perimeter. But points that are more than 1 unit away from the perimeter would not be included. So S is the set of points that are either inside the nonagon or within 1 unit of the perimeter. But if you take the union, then it's equivalent to the original nonagon expanded by 1 unit? Wait, no. Because expanding the nonagon by 1 unit would include all points within 1 unit of the original nonagon. But the original nonagon is already included in that. So maybe S is the Minkowski sum of the nonagon with a closed disk of radius 1? But then the perimeter of S would be the original perimeter plus something?

Wait, maybe I need to visualize this. Let me think. If you have a shape and you take all points within distance 1 from it, then the perimeter of that new shape would be the original perimeter plus twice the length of the original perimeter times something? Wait, no. Wait, when you take all points within distance r from a shape, the perimeter of the new shape is the original perimeter plus 2πr? Wait, that's if the shape is a circle. For a polygon, maybe it's different. Wait, actually, when you take the offset (or Minkowski sum) of a polygon with a disk of radius r, the perimeter becomes the original perimeter plus 2πr. Is that true? Wait, let me think. If you have a polygon, and you take all points within distance r of it, then the perimeter of the resulting shape would be the original perimeter plus the circumference of a circle with radius r? Wait, that seems too simple. Let me think of a square. Suppose you have a square with side length a. If you take all points within distance r of the square, then the perimeter of the resulting shape would be the original perimeter (which is 4a) plus the circumference of a circle with radius r? Wait, no. Because when you offset a square, the corners become rounded with quarter-circles. So the total perimeter would be the original perimeter plus the circumference of a full circle. Wait, because each corner contributes a quarter-circle, and there are four corners, so 4*(1/4)*2πr = 2πr. So the total perimeter would be original perimeter + 2πr. So yes, for a polygon, the perimeter of the offset shape is original perimeter + 2πr. So if that's the case, then in this problem, S is the set of points within 1 unit of the nonagon, which would have a perimeter equal to the original perimeter plus 2π*1. But wait, the problem says S is the union of the interior of the nonagon and the points within 1 unit of the perimeter. But isn't the set of points within 1 unit of the perimeter already including the interior? Wait, no. Wait, if you take all points within 1 unit of the perimeter, that includes points inside the nonagon that are within 1 unit of the perimeter. But the original interior is all points inside the nonagon. So if the nonagon has a certain radius, then points near the center are more than 1 unit away from the perimeter. Therefore, the union of the interior and the points within 1 unit of the perimeter is actually the same as the set of points within 1 unit of the perimeter union the interior. But since the interior is already included in the points within 1 unit of the perimeter? Wait, no. Wait, if you take all points within 1 unit of the perimeter, that includes points inside the nonagon that are within 1 unit of the perimeter. But the original interior is all points inside the nonagon. So if the nonagon is, say, a square with side length 2, then the distance from the center to the perimeter is sqrt(2) ≈ 1.414. So if you take all points within 1 unit of the perimeter, that would include the entire square except for a smaller square in the center. Therefore, the union of the interior and the points within 1 unit of the perimeter would be the entire original square plus the points within 1 unit of the perimeter. But since the original square is already included in the points within 1 unit of the perimeter? Wait, no. Wait, the points within 1 unit of the perimeter would include the perimeter itself and the area inside the square that is within 1 unit of the perimeter. But the original interior is all points inside the square. So if the square is of side length 2, then the distance from the center to the perimeter is sqrt(2) ≈ 1.414, so points within 1 unit of the perimeter would include all points in the square except for a smaller square of side length 2 - 2*1 = 0? Wait, no. Wait, if you have a square of side length 2, then the distance from the center to the perimeter is sqrt(2). If you take all points within 1 unit of the perimeter, that would be the area between the original square and a smaller square inset by 1 unit. But if the original square has side length 2, then the inset square would have side length 2 - 2*1 = 0? Wait, that can't be. Wait, if you move inward by 1 unit from each side, then the side length would be 2 - 2*1 = 0. So that would mean that the area within 1 unit of the perimeter is the entire original square? Wait, that can't be. Wait, if the square has side length 2, then the distance from the center to the side is 1 unit. Wait, no. Wait, if the square has side length 2, then the distance from the center to the side is half the side length divided by tan(π/4). Wait, no. Wait, for a square, the distance from the center to the side is half the side length divided by sqrt(2)? Wait, no. Wait, the distance from the center to the side is half the side length. Wait, if the square has side length 2, then the distance from the center to each side is 1 unit. So if you take all points within 1 unit of the perimeter, that would include all points in the square, because the distance from the center to the perimeter is 1 unit. Therefore, the set of points within 1 unit of the perimeter would be the entire square. Therefore, the union of the interior and the points within 1 unit of the perimeter would just be the interior. Wait, but that seems contradictory. Wait, maybe my example is not correct. Let me think again.

Suppose the square has side length 2. The distance from the center to the perimeter is 1 unit. Therefore, any point inside the square is at most 1 unit away from the perimeter. Therefore, the set of points within 1 unit of the perimeter is the entire square. Therefore, the union of the interior and the points within 1 unit of the perimeter is just the interior. Therefore, in this case, S would be the interior of the square. But that seems odd. Wait, but the problem says "the union of the set of all points inside a regular nonagon... and the set of all points less than 1 unit away from a point on the perimeter of the nonagon." So if the nonagon has a certain radius, then points inside the nonagon are already within some distance from the perimeter. So if the nonagon is such that the distance from the center to the perimeter is more than 1 unit, then the set of points within 1 unit of the perimeter would not include the entire interior. Therefore, in that case, the union would be the original nonagon plus the buffer around it. But if the distance from the center to the perimeter is less than 1 unit, then the union would just be the original nonagon. Therefore, we need to figure out the distance from the center to the perimeter of the nonagon to see whether the buffer overlaps with the interior or not.

So first, let me compute the distance from the center to the perimeter of the regular nonagon. Wait, that's the radius of the nonagon. For a regular nonagon (9-gon), the radius (distance from center to a vertex) is related to the side length. Let me recall the formula. For a regular polygon with n sides of length s, the radius R is given by R = s / (2 sin(π/n)). So here, n = 9, s = 2. Therefore, R = 2 / (2 sin(π/9)) = 1 / sin(π/9). Let me compute sin(π/9). π/9 radians is 20 degrees. The sine of 20 degrees is approximately 0.3420. Therefore, R ≈ 1 / 0.3420 ≈ 2.9238 units. Therefore, the distance from the center to the perimeter (the radius) is approximately 2.92 units. Therefore, since 1 unit is much less than that, the set of points within 1 unit of the perimeter would not include the entire interior. Therefore, the union of the interior and the points within 1 unit of the perimeter is the original nonagon plus a 1-unit wide buffer around it. Therefore, S is like the original nonagon expanded outward by 1 unit. Therefore, the perimeter of S would be the original perimeter plus the circumference of a circle with radius 1? Wait, but earlier with the square example, we saw that the perimeter of the offset shape is original perimeter + 2πr. So maybe that formula applies here as well? If that's the case, then the perimeter of S would be the original perimeter of the nonagon plus 2π*1. Therefore, we need to compute the original perimeter of the nonagon and add 2π.

But let me verify this with the square example. Suppose we have a square with side length 2, so original perimeter is 8. If we take all points within 1 unit of the square, then the perimeter of that shape would be original perimeter + 2π*1 = 8 + 2π. But when we think about the offset square, the perimeter is the original perimeter (which is 8) plus the circumference of a circle with radius 1? Wait, but when you offset a square, the corners become rounded with quarter-circles, so total added circumference is 4*(1/4)*2πr = 2πr. Therefore, yes, the perimeter becomes original perimeter + 2πr. Therefore, in this case, if S is the set of points within 1 unit of the nonagon, then its perimeter is original perimeter + 2π*1. However, in our problem, S is the union of the interior of the nonagon and the points within 1 unit of the perimeter. But if the original nonagon is entirely within the set of points within 1 unit of its perimeter? Wait, no. Wait, if the distance from the center to the perimeter is about 2.92 units, then points inside the nonagon are at most 2.92 units from the center, but the distance from a point inside the nonagon to the perimeter depends on where it is. For example, the center is 2.92 units away from the perimeter. So points near the center are more than 1 unit away from the perimeter. Therefore, the set of points within 1 unit of the perimeter does not include the entire interior. Therefore, the union of the interior and the points within 1 unit of the perimeter is the original nonagon plus the buffer around it. Therefore, S is the Minkowski sum of the nonagon with a disk of radius 1. Therefore, the perimeter of S is original perimeter + 2π*1. Therefore, the answer would be original perimeter of nonagon + 2π. Therefore, I need to compute the original perimeter of the nonagon.

Original perimeter of a regular nonagon with side length 2 is 9*2 = 18 units. Therefore, perimeter of S would be 18 + 2π. But wait, let me check again. Wait, if S is the union of the interior and the points within 1 unit of the perimeter, then is that equivalent to the Minkowski sum? Because the Minkowski sum would include all points within 1 unit of the nonagon, which includes the interior and the buffer. Therefore, yes, S is exactly the Minkowski sum. Therefore, the perimeter of S is original perimeter + 2π*1. Therefore, 18 + 2π. But let me think again. Wait, in the square example, if you take the Minkowski sum of the square with a disk of radius r, then the perimeter is original perimeter + 2πr. But in that case, the Minkowski sum includes the original square and the buffer. However, in our problem, S is the union of the interior of the nonagon and the points within 1 unit of the perimeter. But the points within 1 unit of the perimeter include the perimeter itself and the area inside the nonagon that is within 1 unit of the perimeter. Therefore, the union of the interior and the points within 1 unit of the perimeter is the same as the set of points within 1 unit of the nonagon. Wait, no. Wait, the set of points within 1 unit of the nonagon includes all points that are within 1 unit of any point on the nonagon. Which includes the interior of the nonagon (since every point in the interior is within 1 unit of the perimeter, but not necessarily within 1 unit of the entire nonagon? Wait, no. Wait, if you take a point inside the nonagon, then it is within 1 unit of the perimeter, but is it within 1 unit of the nonagon? Well, the nonagon is a closed set, so the distance from the point to the nonagon is zero, since it's inside. Wait, but the set of points within 1 unit of the nonagon would include all points that are within 1 unit of any point on the nonagon. Which includes the interior of the nonagon, because any interior point is within 0 units of the nonagon. Therefore, the set of points within 1 unit of the nonagon is the same as the union of the interior of the nonagon and the points within 1 unit of the perimeter. Wait, no. Wait, the set of points within 1 unit of the nonagon is the same as the Minkowski sum of the nonagon with a disk of radius 1. Which includes the original nonagon and the buffer around it. Therefore, the union of the interior of the nonagon and the points within 1 unit of the perimeter is exactly the same as the set of points within 1 unit of the nonagon. Therefore, S is the Minkowski sum, and its perimeter is original perimeter + 2π*1. Therefore, the answer is 18 + 2π. But let me check again with the square example. Suppose the square has side length 2, original perimeter 8. If we take all points within 1 unit of the square, then the perimeter of that shape is 8 + 2π*1. But if we compute it manually, the offset square would have its sides extended outward by 1 unit, but with rounded corners. Each side of the square is length 2, so when you offset outward by 1 unit, the length of each side remains 2, but the corners are replaced by quarter-circles of radius 1. Therefore, the total perimeter is 4*2 + 4*(1/4)*2π*1 = 8 + 2π. Which matches the formula. Therefore, the formula seems correct. Therefore, in our problem, the perimeter of S is original perimeter + 2π*1. Therefore, since original perimeter is 9*2 = 18, then perimeter of S is 18 + 2π. Therefore, the answer is 18 + 2π. But wait, the problem says "the set of all points inside a regular nonagon... and the set of all points less than 1 unit away from a point on the perimeter of the nonagon." So maybe there is an overlap? But as we discussed, the set of points less than 1 unit away from the perimeter includes the interior points that are within 1 unit of the perimeter, but not the entire interior. However, the union of the interior and the points within 1 unit of the perimeter is the same as the set of points within 1 unit of the nonagon. Therefore, the perimeter of S is 18 + 2π. However, let me think again. Wait, if you take the union of the interior of the nonagon and the points within 1 unit of the perimeter, then the perimeter of S would be the perimeter of the buffer zone, but not sure. Wait, maybe not. Wait, when you take the union, the perimeter of S would be the perimeter of the buffer zone, but since the original nonagon is inside the buffer zone, the perimeter of S would be the same as the perimeter of the buffer zone. But the buffer zone is the Minkowski sum, which has perimeter original perimeter + 2π*1. Therefore, the answer is 18 + 2π. However, let me check if there is a different interpretation. Maybe the problem is not about the Minkowski sum, but something else. For example, if you take the interior of the nonagon and add all points within 1 unit of the perimeter, then maybe the perimeter of S is the original perimeter plus twice the length of the perimeter? No, that doesn't make sense. Wait, no. Wait, if you have the original nonagon, and then you add a 1-unit buffer around it, then the perimeter of the new shape would be the original perimeter plus the circumference of a circle with radius 1? Wait, but in the square example, it was original perimeter + 2πr. So maybe that's the case here. Therefore, the answer is 18 + 2π. However, let me think again. Wait, the perimeter of the buffer zone. When you take the buffer around the nonagon, the perimeter would consist of the original edges extended outward by 1 unit, but with rounded corners. Each original edge is straight, and when you offset it outward by 1 unit, the length of each edge remains the same, but the corners are replaced by circular arcs. Therefore, the total perimeter would be the original perimeter plus the circumference of a circle with radius 1. Wait, but in the square example, the original perimeter is 8, and the buffer perimeter is 8 + 2π. So in general, for any polygon, the perimeter of the offset shape is original perimeter + 2πr. Therefore, in this case, the perimeter of S is 18 + 2π. Therefore, the answer is 18 + 2π. However, let me check if there is a mistake here. For example, if the original nonagon has a perimeter of 18, and we add a buffer of 1 unit, then the perimeter of S is 18 + 2π. But maybe there is something else? Wait, but in the square example, the buffer adds 2πr to the perimeter. So why is that? Because when you offset the polygon, each corner is replaced by a circular arc. For a polygon with n sides, each corner is replaced by a circular arc of angle equal to the exterior angle. For a regular polygon, each exterior angle is 2π/n. Therefore, the total angle of all arcs is n*(2π/n) = 2π. Therefore, the total length of all arcs is 2πr. Therefore, the perimeter of the offset shape is original perimeter + 2πr. Therefore, yes, that formula holds. Therefore, in our case, the perimeter of S is 18 + 2π. Therefore, the answer is 18 + 2π. However, let me check if the problem says "less than 1 unit away from a point on the perimeter". So does that include the perimeter itself? If it's "less than 1 unit away", then it's the open disk around the perimeter, but since we are taking the union with the interior of the nonagon, which includes the perimeter. However, when considering the perimeter of S, the boundary of S would be the boundary of the buffer zone. Since the buffer zone is the set of points within 1 unit of the nonagon, which includes the original perimeter and the buffer. However, the perimeter of S would be the boundary of this buffer zone. Which, as we discussed, is the original perimeter offset outward by 1 unit, with rounded corners. Therefore, the perimeter is original perimeter + 2π*1. Therefore, 18 + 2π. Therefore, I think that is the answer. However, let me check if there is a different approach. For example, maybe the perimeter of S is just the perimeter of the buffer zone, which is the original perimeter plus the circumference of the circle? Wait, but in the square example, the buffer zone's perimeter is original perimeter + 2πr. Therefore, same here. Therefore, answer is 18 + 2π. However, let me check if the problem says "the set of all points inside a regular nonagon... and the set of all points less than 1 unit away from a point on the perimeter". So if you take the union, then the perimeter of S would be the boundary of the union. The union is the original nonagon plus the buffer. However, the original nonagon's boundary is entirely inside the buffer zone. Wait, no. The buffer zone is all points within 1 unit of the perimeter. Therefore, the original perimeter is part of the buffer zone. Therefore, the union of the interior of the nonagon and the buffer zone is the same as the buffer zone. Wait, no. Wait, the buffer zone includes all points within 1 unit of the perimeter. The interior of the nonagon includes all points inside the nonagon. However, the buffer zone includes points inside the nonagon that are within 1 unit of the perimeter. Therefore, the union of the interior and the buffer zone is the set of points within 1 unit of the perimeter union the interior. But since the interior is already included in the buffer zone? Wait, no. Wait, if you take all points within 1 unit of the perimeter, that includes points inside the nonagon that are within 1 unit of the perimeter. However, the interior of the nonagon includes all points inside the nonagon, which may be more than 1 unit away from the perimeter. Therefore, the union of the interior and the buffer zone is the set of points within 1 unit of the perimeter union the interior. Which is the same as the set of points within 1 unit of the perimeter union the interior. But since the interior is not entirely within the buffer zone, the union is the buffer zone plus the interior points that are more than 1 unit away from the perimeter. Wait, but that seems contradictory. Wait, if the buffer zone is all points within 1 unit of the perimeter, then the union of the interior and the buffer zone is the set of points that are either inside the nonagon or within 1 unit of the perimeter. But since the interior is all points inside the nonagon, which includes points that are more than 1 unit away from the perimeter. Therefore, the union is the entire nonagon plus the buffer zone. Wait, but the buffer zone is already extending outward from the perimeter. Therefore, the union is the nonagon expanded by 1 unit. Therefore, the perimeter of S is the perimeter of the expanded nonagon, which is original perimeter + 2π*1. Therefore, 18 + 2π. Therefore, I think that is correct. Therefore, the answer is 18 + 2π. However, let me check if there is a different way to think about it. For example, if you imagine the perimeter of S. The perimeter of S would consist of two parts: the original perimeter of the nonagon, but offset outward by 1 unit, and the circular arcs at each corner. Wait, but when you take the union of the interior and the buffer, the perimeter of S would be the boundary of the buffer zone. Because the interior of the nonagon is entirely inside the buffer zone. Wait, no. Wait, the buffer zone is all points within 1 unit of the perimeter. The interior of the nonagon is all points inside the nonagon. So if a point is inside the nonagon and more than 1 unit away from the perimeter, then it is not in the buffer zone. Therefore, the union of the interior and the buffer zone is the buffer zone plus the interior points that are within 1 unit of the perimeter. Wait, but the buffer zone already includes those interior points. Therefore, the union is the buffer zone union the interior points that are within 1 unit of the perimeter. But since the buffer zone already includes those points, the union is just the buffer zone. Wait, no. Wait, the buffer zone is all points within 1 unit of the perimeter. The interior of the nonagon is all points inside the nonagon. Therefore, the union is the set of points that are either inside the nonagon or within 1 unit of the perimeter. Which is the same as the set of points that are within 1 unit of the perimeter or inside the nonagon. But since the interior of the nonagon is not entirely within the buffer zone, the union is the buffer zone plus the interior points that are more than 1 unit away from the perimeter. Wait, but how can that be? If you take a point inside the nonagon that is more than 1 unit away from the perimeter, then it is not in the buffer zone. Therefore, the union is the buffer zone plus those interior points. But then the perimeter of S would be the boundary of the buffer zone plus the boundary of those interior points? Wait, no. The union is a connected region. The buffer zone is a sort of "thickened" perimeter, and the interior points that are more than 1 unit away from the perimeter are the inner part. Wait, but if you take the union of the interior and the buffer zone, then the entire region is the buffer zone union the interior. But since the buffer zone already includes the perimeter and the interior points within 1 unit of the perimeter, the union is the buffer zone union the interior points that are more than 1 unit away from the perimeter. Wait, but how does that look? It's like the original nonagon with a 1-unit wide strip around it. Therefore, the perimeter of S would be the outer boundary of the buffer zone, which is the original perimeter offset outward by 1 unit, plus the inner boundary of the buffer zone? Wait, no. Wait, the buffer zone is all points within 1 unit of the perimeter. So the buffer zone is like a 1-unit wide region around the nonagon. Therefore, the union of the interior and the buffer zone is the original nonagon plus the buffer zone. Therefore, the perimeter of S would be the outer boundary of the buffer zone, which is the original perimeter offset outward by 1 unit, with rounded corners. Therefore, the perimeter is original perimeter + 2π*1. Therefore, 18 + 2π. Therefore, I think that is correct. Therefore, the answer is 18 + 2π. However, let me check if there is a different interpretation. For example, maybe the problem is not about the Minkowski sum, but about the union of the interior and the offset perimeter. But in that case, the perimeter would be the same as the Minkowski sum. Therefore, I think the answer is 18 + 2π. However, let me check with another example. Suppose the nonagon has a very small side length, such that the distance from the center to the perimeter is less than 1. Then the set of points within 1 unit of the perimeter would include the entire nonagon. Therefore, the union of the interior and the points within 1 unit of the perimeter would be the same as the set of points within 1 unit of the perimeter, which is the Minkowski sum. Therefore, the perimeter would be original perimeter + 2π*1. However, if the nonagon is very large, then the same formula applies. Therefore, regardless of the size, the perimeter of S is original perimeter + 2π*1. Therefore, in our case, since the original perimeter is 18, the answer is 18 + 2π. Therefore, I think that is the answer. Therefore, I will go with 18 + 2π. However, let me check if the problem says "less than 1 unit away", which is an open disk, but since we are taking the union with the interior, which is a closed set, the perimeter of S would still be the same as the Minkowski sum. Therefore, the answer is 18 + 2π. Therefore, I think that is correct.

**Final Answer**
\boxed{18 + 2\pi}

To determine the perimeter of the set $ S $, we begin by understanding its definition:

> $ S $ is the union of:
> - The set of all points **inside** a regular nonagon with side length 2 units, and
> - The set of all points **less than 1 unit away** from a point on the perimeter of the nonagon.

---

### Step 1: Understanding the Geometric Structure

This union can be interpreted as the **Minkowski sum** of the nonagon with a **disk of radius 1**. That is, $ S $ is the set of all points that are within 1 unit of the nonagon, including:
- The interior of the nonagon,
- The perimeter of the nonagon,
- A 1-unit wide "buffer" zone around the nonagon.

This is equivalent to **expanding the nonagon outward by 1 unit in all directions**, forming a new shape that includes the original nonagon and a 1-unit wide strip around it.

---

### Step 2: Perimeter of the Expanded Shape

When a polygon is expanded outward by a distance $ r $, the resulting shape's perimeter is given by:

$$
\text{New Perimeter} = \text{Original Perimeter} + 2\pi r
$$

This formula arises because:
- The original edges are extended outward by $ r $, but their lengths remain the same.
- The corners of the polygon are replaced by **circular arcs** of radius $ r $, and the total angle of these arcs is $ 2\pi $, contributing a total arc length of $ 2\pi r $.

---

### Step 3: Apply to the Given Nonagon

- The nonagon is **regular** with **side length 2**.
- It has **9 sides**, so its **original perimeter** is:

$$
P = 9 \times 2 = 18
$$

- The buffer zone is **1 unit wide**, so $ r = 1 $.

Thus, the perimeter of the expanded shape (i.e., the perimeter of $ S $) is:

$$
P_S = 18 + 2\pi
$$

---

### Final Answer

$$
\boxed{18 + 2\pi}
$$

[Assistant]
Explanation:
Okay, so I need to find the perimeter of this set S. Let me try to understand what S is. It says S is the union of two things: first, all the points inside a regular nonagon with side length 2 units, and second, all the points less than 1 unit away from a point on the perimeter of the nonagon. So, essentially, S is the original nonagon plus a sort of "buffer" zone around it, right? Like, if you imagine taking the nonagon and then expanding it outward by 1 unit in all directions, but only up to the points that are within 1 unit of the perimeter. Wait, but actually, since it's the union of the interior of the nonagon and all points within 1 unit of the perimeter, that would be like taking the original nonagon and then adding a 1-unit wide strip around it. But actually, when you take all points within 1 unit of the perimeter, that's like taking the Minkowski sum of the perimeter with a disk of radius 1. But since the original nonagon is already included, maybe S is the original nonagon expanded by 1 unit? Hmm, but maybe not exactly, because when you take all points within 1 unit of the perimeter, that includes the original nonagon? Wait, no. If you take all points within 1 unit of the perimeter, that would include the original perimeter and the area inside the nonagon? Wait, no. Wait, if you take all points less than 1 unit away from the perimeter, that would include points inside the nonagon? Let me think.

Suppose you have a point inside the nonagon. Is it less than 1 unit away from the perimeter? Well, if the nonagon has a certain radius, then points near the center would be more than 1 unit away from the perimeter. So maybe S is the union of the interior of the nonagon and the area within 1 unit of the perimeter. So that would be like the original nonagon expanded outward by 1 unit, but also including the area inside the original nonagon? Wait, but if you take the union of the interior and the points within 1 unit of the perimeter, then the interior is already included in the points within 1 unit of the perimeter? Wait, no. If you take all points within 1 unit of the perimeter, that would include points inside the nonagon that are within 1 unit of the perimeter. But points that are more than 1 unit away from the perimeter would not be included. So S is the set of points that are either inside the nonagon or within 1 unit of the perimeter. But if you take the union, then it's equivalent to the original nonagon expanded by 1 unit? Wait, no. Because expanding the nonagon by 1 unit would include all points within 1 unit of the original nonagon. But the original nonagon is already included in that. So maybe S is the Minkowski sum of the nonagon with a closed disk of radius 1? But then the perimeter of S would be the original perimeter plus something?

Wait, maybe I need to visualize this. Let me think. If you have a shape and you take all points within distance 1 from it, then the perimeter of that new shape would be the original perimeter plus twice the length of the original perimeter times something? Wait, no. Wait, when you take all points within distance r from a shape, the perimeter of the new shape is the original perimeter plus 2πr? Wait, that's if the shape is a circle. For a polygon, maybe it's different. Wait, actually, when you take the offset (or Minkowski sum) of a polygon with a disk of radius r, the perimeter becomes the original perimeter plus 2πr. Is that true? Wait, let me think. If you have a polygon, and you take all points within distance r of it, then the perimeter of the resulting shape would be the original perimeter plus the circumference of a circle with radius r? Wait, that seems too simple. Let me think of a square. Suppose you have a square with side length a. If you take all points within distance r of the square, then the perimeter of the resulting shape would be the original perimeter (which is 4a) plus the circumference of a circle with radius r? Wait, no. Because when you offset a square, the corners become rounded with quarter-circles. So the total perimeter would be the original perimeter plus the circumference of a full circle. Wait, because each corner contributes a quarter-circle, and there are four corners, so 4*(1/4)*2πr = 2πr. So the total perimeter would be original perimeter + 2πr. So yes, for a polygon, the perimeter of the offset shape is original perimeter + 2πr. So if that's the case, then in this problem, S is the set of points within 1 unit of the nonagon, which would have a perimeter equal to the original perimeter plus 2π*1. But wait, the problem says S is the union of the interior of the nonagon and the points within 1 unit of the perimeter. But isn't the set of points within 1 unit of the perimeter already including the interior? Wait, no. Wait, if you take all points within 1 unit of the perimeter, that includes points inside the nonagon that are within 1 unit of the perimeter. But the original interior is all points inside the nonagon. So if the nonagon has a certain radius, then points near the center are more than 1 unit away from the perimeter. Therefore, the union of the interior and the points within 1 unit of the perimeter is actually the same as the set of points within 1 unit of the perimeter union the interior. But since the interior is already included in the points within 1 unit of the perimeter? Wait, no. Wait, if you take all points within 1 unit of the perimeter, that includes points inside the nonagon that are within 1 unit of the perimeter. But the original interior is all points inside the nonagon. So if the nonagon is, say, a square with side length 2, then the distance from the center to the perimeter is sqrt(2) ≈ 1.414. So if you take all points within 1 unit of the perimeter, that would include the entire square except for a smaller square in the center. Therefore, the union of the interior and the points within 1 unit of the perimeter would be the entire original square plus the points within 1 unit of the perimeter. But since the original square is already included in the points within 1 unit of the perimeter? Wait, no. Wait, the points within 1 unit of the perimeter would include the perimeter itself and the area inside the square that is within 1 unit of the perimeter. But the original interior is all points inside the square. So if the square is of side length 2, then the distance from the center to the perimeter is sqrt(2) ≈ 1.414, so points within 1 unit of the perimeter would include all points in the square except for a smaller square of side length 2 - 2*1 = 0? Wait, no. Wait, if you have a square of side length 2, then the distance from the center to the perimeter is sqrt(2). If you take all points within 1 unit of the perimeter, that would be the area between the original square and a smaller square inset by 1 unit. But if the original square has side length 2, then the inset square would have side length 2 - 2*1 = 0? Wait, that can't be. Wait, if you move inward by 1 unit from each side, then the side length would be 2 - 2*1 = 0. So that would mean that the area within 1 unit of the perimeter is the entire original square? Wait, that can't be. Wait, if the square has side length 2, then the distance from the center to the side is 1 unit. Wait, no. Wait, if the square has side length 2, then the distance from the center to the side is half the side length divided by tan(π/4). Wait, no. Wait, for a square, the distance from the center to the side is half the side length divided by sqrt(2)? Wait, no. Wait, the distance from the center to the side is half the side length. Wait, if the square has side length 2, then the distance from the center to each side is 1 unit. So if you take all points within 1 unit of the perimeter, that would include all points in the square, because the distance from the center to the perimeter is 1 unit. Therefore, the set of points within 1 unit of the perimeter would be the entire square. Therefore, the union of the interior and the points within 1 unit of the perimeter would just be the interior. Wait, but that seems contradictory. Wait, maybe my example is not correct. Let me think again.

Suppose the square has side length 2. The distance from the center to the perimeter is 1 unit. Therefore, any point inside the square is at most 1 unit away from the perimeter. Therefore, the set of points within 1 unit of the perimeter is the entire square. Therefore, the union of the interior and the points within 1 unit of the perimeter is just the interior. Therefore, in this case, S would be the interior of the square. But that seems odd. Wait, but the problem says "the union of the set of all points inside a regular nonagon... and the set of all points less than 1 unit away from a point on the perimeter of the nonagon." So if the nonagon has a certain radius, then points inside the nonagon are already within some distance from the perimeter. So if the nonagon is such that the distance from the center to the perimeter is more than 1 unit, then the set of points within 1 unit of the perimeter would not include the entire interior. Therefore, in that case, the union would be the original nonagon plus the buffer around it. But if the distance from the center to the perimeter is less than 1 unit, then the union would just be the original nonagon. Therefore, we need to figure out the distance from the center to the perimeter of the nonagon to see whether the buffer overlaps with the interior or not.

So first, let me compute the distance from the center to the perimeter of the regular nonagon. Wait, that's the radius of the nonagon. For a regular nonagon (9-gon), the radius (distance from center to a vertex) is related to the side length. Let me recall the formula. For a regular polygon with n sides of length s, the radius R is given by R = s / (2 sin(π/n)). So here, n = 9, s = 2. Therefore, R = 2 / (2 sin(π/9)) = 1 / sin(π/9). Let me compute sin(π/9). π/9 radians is 20 degrees. The sine of 20 degrees is approximately 0.3420. Therefore, R ≈ 1 / 0.3420 ≈ 2.9238 units. Therefore, the distance from the center to the perimeter (the radius) is approximately 2.92 units. Therefore, since 1 unit is much less than that, the set of points within 1 unit of the perimeter would not include the entire interior. Therefore, the union of the interior and the points within 1 unit of the perimeter is the original nonagon plus a 1-unit wide buffer around it. Therefore, S is like the original nonagon expanded outward by 1 unit. Therefore, the perimeter of S would be the original perimeter plus the circumference of a circle with radius 1? Wait, but earlier with the square example, we saw that the perimeter of the offset shape is original perimeter + 2πr. So maybe that formula applies here as well? If that's the case, then the perimeter of S would be the original perimeter of the nonagon plus 2π*1. Therefore, we need to compute the original perimeter of the nonagon and add 2π.

But let me verify this with the square example. Suppose we have a square with side length 2, so original perimeter is 8. If we take all points within 1 unit of the square, then the perimeter of that shape would be original perimeter + 2π*1 = 8 + 2π. But when we think about the offset square, the perimeter is the original perimeter (which is 8) plus the circumference of a circle with radius 1? Wait, but when you offset a square, the corners become rounded with quarter-circles, so total added circumference is 4*(1/4)*2πr = 2πr. Therefore, yes, the perimeter becomes original perimeter + 2πr. Therefore, in this case, if S is the set of points within 1 unit of the nonagon, then its perimeter is original perimeter + 2π*1. However, in our problem, S is the union of the interior of the nonagon and the points within 1 unit of the perimeter. But if the original nonagon is entirely within the set of points within 1 unit of its perimeter? Wait, no. Wait, if the distance from the center to the perimeter is about 2.92 units, then points inside the nonagon are at most 2.92 units from the center, but the distance from a point inside the nonagon to the perimeter depends on where it is. For example, the center is 2.92 units away from the perimeter. So points near the center are more than 1 unit away from the perimeter. Therefore, the set of points within 1 unit of the perimeter does not include the entire interior. Therefore, the union of the interior and the points within 1 unit of the perimeter is the original nonagon plus the buffer around it. Therefore, S is the Minkowski sum of the nonagon with a disk of radius 1. Therefore, the perimeter of S is original perimeter + 2π*1. Therefore, the answer would be original perimeter of nonagon + 2π. Therefore, I need to compute the original perimeter of the nonagon.

Original perimeter of a regular nonagon with side length 2 is 9*2 = 18 units. Therefore, perimeter of S would be 18 + 2π. But wait, let me check again. Wait, if S is the union of the interior and the points within 1 unit of the perimeter, then is that equivalent to the Minkowski sum? Because the Minkowski sum would include all points within 1 unit of the nonagon, which includes the interior and the buffer. Therefore, yes, S is exactly the Minkowski sum. Therefore, the perimeter of S is original perimeter + 2π*1. Therefore, 18 + 2π. But let me think again. Wait, in the square example, if you take the Minkowski sum of the square with a disk of radius r, then the perimeter is original perimeter + 2πr. But in that case, the Minkowski sum includes the original square and the buffer. However, in our problem, S is the union of the interior of the nonagon and the points within 1 unit of the perimeter. But the points within 1 unit of the perimeter include the perimeter itself and the area inside the nonagon that is within 1 unit of the perimeter. Therefore, the union of the interior and the points within 1 unit of the perimeter is the same as the set of points within 1 unit of the nonagon. Wait, no. Wait, the set of points within 1 unit of the nonagon includes all points that are within 1 unit of any point on the nonagon. Which includes the interior of the nonagon (since every point in the interior is within 1 unit of the perimeter, but not necessarily within 1 unit of the entire nonagon? Wait, no. Wait, if you take a point inside the nonagon, then it is within 1 unit of the perimeter, but is it within 1 unit of the nonagon? Well, the nonagon is a closed set, so the distance from the point to the nonagon is zero, since it's inside. Wait, but the set of points within 1 unit of the nonagon would include all points that are within 1 unit of any point on the nonagon. Which includes the interior of the nonagon, because any interior point is within 0 units of the nonagon. Therefore, the set of points within 1 unit of the nonagon is the same as the union of the interior of the nonagon and the points within 1 unit of the perimeter. Wait, no. Wait, the set of points within 1 unit of the nonagon is the same as the Minkowski sum of the nonagon with a disk of radius 1. Which includes the original nonagon and the buffer around it. Therefore, the union of the interior of the nonagon and the points within 1 unit of the perimeter is exactly the same as the set of points within 1 unit of the nonagon. Therefore, S is the Minkowski sum, and its perimeter is original perimeter + 2π*1. Therefore, the answer is 18 + 2π. But let me check again with the square example. Suppose the square has side length 2, original perimeter 8. If we take all points within 1 unit of the square, then the perimeter of that shape is 8 + 2π*1. But if we compute it manually, the offset square would have its sides extended outward by 1 unit, but with rounded corners. Each side of the square is length 2, so when you offset outward by 1 unit, the length of each side remains 2, but the corners are replaced by quarter-circles of radius 1. Therefore, the total perimeter is 4*2 + 4*(1/4)*2π*1 = 8 + 2π. Which matches the formula. Therefore, the formula seems correct. Therefore, in our problem, the perimeter of S is original perimeter + 2π*1. Therefore, since original perimeter is 9*2 = 18, then perimeter of S is 18 + 2π. Therefore, the answer is 18 + 2π. But wait, the problem says "the set of all points inside a regular nonagon... and the set of all points less than 1 unit away from a point on the perimeter of the nonagon." So maybe there is an overlap? But as we discussed, the set of points less than 1 unit away from the perimeter includes the interior points that are within 1 unit of the perimeter, but not the entire interior. However, the union of the interior and the points within 1 unit of the perimeter is the same as the set of points within 1 unit of the nonagon. Therefore, the perimeter of S is 18 + 2π. However, let me think again. Wait, if you take the union of the interior of the nonagon and the points within 1 unit of the perimeter, then the perimeter of S would be the perimeter of the buffer zone, but not sure. Wait, maybe not. Wait, when you take the union, the perimeter of S would be the perimeter of the buffer zone, but since the original nonagon is inside the buffer zone, the perimeter of S would be the same as the perimeter of the buffer zone. But the buffer zone is the Minkowski sum, which has perimeter original perimeter + 2π*1. Therefore, the answer is 18 + 2π. However, let me check if there is a different interpretation. Maybe the problem is not about the Minkowski sum, but something else. For example, if you take the interior of the nonagon and add all points within 1 unit of the perimeter, then maybe the perimeter of S is the original perimeter plus twice the length of the perimeter? No, that doesn't make sense. Wait, no. Wait, if you have the original nonagon, and then you add a 1-unit buffer around it, then the perimeter of the new shape would be the original perimeter plus the circumference of a circle with radius 1? Wait, but in the square example, it was original perimeter + 2πr. So maybe that's the case here. Therefore, the answer is 18 + 2π. However, let me think again. Wait, the perimeter of the buffer zone. When you take the buffer around the nonagon, the perimeter would consist of the original edges extended outward by 1 unit, but with rounded corners. Each original edge is straight, and when you offset it outward by 1 unit, the length of each edge remains the same, but the corners are replaced by circular arcs. Therefore, the total perimeter would be the original perimeter plus the circumference of a circle with radius 1. Wait, but in the square example, the original perimeter is 8, and the buffer perimeter is 8 + 2π. So in general, for any polygon, the perimeter of the offset shape is original perimeter + 2πr. Therefore, in this case, the perimeter of S is 18 + 2π. Therefore, the answer is 18 + 2π. However, let me check if there is a mistake here. For example, if the original nonagon has a perimeter of 18, and we add a buffer of 1 unit, then the perimeter of S is 18 + 2π. But maybe there is something else? Wait, but in the square example, the buffer adds 2πr to the perimeter. So why is that? Because when you offset the polygon, each corner is replaced by a circular arc. For a polygon with n sides, each corner is replaced by a circular arc of angle equal to the exterior angle. For a regular polygon, each exterior angle is 2π/n. Therefore, the total angle of all arcs is n*(2π/n) = 2π. Therefore, the total length of all arcs is 2πr. Therefore, the perimeter of the offset shape is original perimeter + 2πr. Therefore, yes, that formula holds. Therefore, in our case, the perimeter of S is 18 + 2π. Therefore, the answer is 18 + 2π. However, let me check if the problem says "less than 1 unit away from a point on the perimeter". So does that include the perimeter itself? If it's "less than 1 unit away", then it's the open disk around the perimeter, but since we are taking the union with the interior of the nonagon, which includes the perimeter. However, when considering the perimeter of S, the boundary of S would be the boundary of the buffer zone. Since the buffer zone is the set of points within 1 unit of the nonagon, which includes the original perimeter and the buffer. However, the perimeter of S would be the boundary of this buffer zone. Which, as we discussed, is the original perimeter offset outward by 1 unit, with rounded corners. Therefore, the perimeter is original perimeter + 2π*1. Therefore, 18 + 2π. Therefore, I think that is the answer. However, let me check if there is a different approach. For example, maybe the perimeter of S is just the perimeter of the buffer zone, which is the original perimeter plus the circumference of the circle? Wait, but in the square example, the buffer zone's perimeter is original perimeter + 2πr. Therefore, same here. Therefore, answer is 18 + 2π. However, let me check if the problem says "the set of all points inside a regular nonagon... and the set of all points less than 1 unit away from a point on the perimeter". So if you take the union, then the perimeter of S would be the boundary of the union. The union is the original nonagon plus the buffer. However, the original nonagon's boundary is entirely inside the buffer zone. Wait, no. The buffer zone is all points within 1 unit of the perimeter. Therefore, the original perimeter is part of the buffer zone. Therefore, the union of the interior of the nonagon and the buffer zone is the same as the buffer zone. Wait, no. Wait, the buffer zone includes all points within 1 unit of the perimeter. The interior of the nonagon includes all points inside the nonagon. However, the buffer zone includes points inside the nonagon that are within 1 unit of the perimeter. Therefore, the union of the interior and the buffer zone is the set of points within 1 unit of the perimeter union the interior. But since the interior is already included in the buffer zone? Wait, no. Wait, if you take all points within 1 unit of the perimeter, that includes points inside the nonagon that are within 1 unit of the perimeter. However, the interior of the nonagon includes all points inside the nonagon, which may be more than 1 unit away from the perimeter. Therefore, the union of the interior and the buffer zone is the set of points within 1 unit of the perimeter union the interior. Which is the same as the set of points within 1 unit of the perimeter union the interior. But since the interior is not entirely within the buffer zone, the union is the buffer zone plus the interior points that are more than 1 unit away from the perimeter. Wait, but that seems contradictory. Wait, if the buffer zone is all points within 1 unit of the perimeter, then the union of the interior and the buffer zone is the set of points that are either inside the nonagon or within 1 unit of the perimeter. But since the interior is all points inside the nonagon, which includes points that are more than 1 unit away from the perimeter. Therefore, the union is the entire nonagon plus the buffer zone. Wait, but the buffer zone is already extending outward from the perimeter. Therefore, the union is the nonagon expanded by 1 unit. Therefore, the perimeter of S is the perimeter of the expanded nonagon, which is original perimeter + 2π*1. Therefore, 18 + 2π. Therefore, I think that is correct. Therefore, the answer is 18 + 2π. However, let me check if there is a different way to think about it. For example, if you imagine the perimeter of S. The perimeter of S would consist of two parts: the original perimeter of the nonagon, but offset outward by 1 unit, and the circular arcs at each corner. Wait, but when you take the union of the interior and the buffer, the perimeter of S would be the boundary of the buffer zone. Because the interior of the nonagon is entirely inside the buffer zone. Wait, no. Wait, the buffer zone is all points within 1 unit of the perimeter. The interior of the nonagon is all points inside the nonagon. So if a point is inside the nonagon and more than 1 unit away from the perimeter, then it is not in the buffer zone. Therefore, the union of the interior and the buffer zone is the buffer zone plus the interior points that are within 1 unit of the perimeter. Wait, but the buffer zone already includes those interior points. Therefore, the union is the buffer zone union the interior points that are within 1 unit of the perimeter. But since the buffer zone already includes those points, the union is just the buffer zone. Wait, no. Wait, the buffer zone is all points within 1 unit of the perimeter. The interior of the nonagon is all points inside the nonagon. Therefore, the union is the set of points that are either inside the nonagon or within 1 unit of the perimeter. Which is the same as the set of points that are within 1 unit of the perimeter or inside the nonagon. But since the interior of the nonagon is not entirely within the buffer zone, the union is the buffer zone plus the interior points that are more than 1 unit away from the perimeter. Wait, but how can that be? If you take a point inside the nonagon that is more than 1 unit away from the perimeter, then it is not in the buffer zone. Therefore, the union is the buffer zone plus those interior points. But then the perimeter of S would be the boundary of the buffer zone plus the boundary of those interior points? Wait, no. The union is a connected region. The buffer zone is a sort of "thickened" perimeter, and the interior points that are more than 1 unit away from the perimeter are the inner part. Wait, but if you take the union of the interior and the buffer zone, then the entire region is the buffer zone union the interior. But since the buffer zone already includes the perimeter and the interior points within 1 unit of the perimeter, the union is the buffer zone union the interior points that are more than 1 unit away from the perimeter. Wait, but how does that look? It's like the original nonagon with a 1-unit wide strip around it. Therefore, the perimeter of S would be the outer boundary of the buffer zone, which is the original perimeter offset outward by 1 unit, plus the inner boundary of the buffer zone? Wait, no. Wait, the buffer zone is all points within 1 unit of the perimeter. So the buffer zone is like a 1-unit wide region around the nonagon. Therefore, the union of the interior and the buffer zone is the original nonagon plus the buffer zone. Therefore, the perimeter of S would be the outer boundary of the buffer zone, which is the original perimeter offset outward by 1 unit, with rounded corners. Therefore, the perimeter is original perimeter + 2π*1. Therefore, 18 + 2π. Therefore, I think that is correct. Therefore, the answer is 18 + 2π. However, let me check if there is a different interpretation. For example, maybe the problem is not about the Minkowski sum, but about the union of the interior and the offset perimeter. But in that case, the perimeter would be the same as the Minkowski sum. Therefore, I think the answer is 18 + 2π. However, let me check with another example. Suppose the nonagon has a very small side length, such that the distance from the center to the perimeter is less than 1. Then the set of points within 1 unit of the perimeter would include the entire nonagon. Therefore, the union of the interior and the points within 1 unit of the perimeter would be the same as the set of points within 1 unit of the perimeter, which is the Minkowski sum. Therefore, the perimeter would be original perimeter + 2π*1. However, if the nonagon is very large, then the same formula applies. Therefore, regardless of the size, the perimeter of S is original perimeter + 2π*1. Therefore, in our case, since the original perimeter is 18, the answer is 18 + 2π. Therefore, I think that is the answer. Therefore, I will go with 18 + 2π. However, let me check if the problem says "less than 1 unit away", which is an open disk, but since we are taking the union with the interior, which is a closed set, the perimeter of S would still be the same as the Minkowski sum. Therefore, the answer is 18 + 2π. Therefore, I think that is correct.

**Final Answer**
\boxed{18 + 2\pi}

To determine the perimeter of the set $ S $, we begin by understanding its definition:

> $ S $ is the union of:
> - The set of all points **inside** a regular nonagon with side length 2 units, and
> - The set of all points **less than 1 unit away** from a point on the perimeter of the nonagon.

---

### Step 1: Understanding the Geometric Structure

This union can be interpreted as the **Minkowski sum** of the nonagon with a **disk of radius 1**. That is, $ S $ is the set of all points that are within 1 unit of the nonagon, including:
- The interior of the nonagon,
- The perimeter of the nonagon,
- A 1-unit wide "buffer" zone around the nonagon.

This is equivalent to **expanding the nonagon outward by 1 unit in all directions**, forming a new shape that includes the original nonagon and a 1-unit wide strip around it.

---

### Step 2: Perimeter of the Expanded Shape

When a polygon is expanded outward by a distance $ r $, the resulting shape's perimeter is given by:

$$
\text{New Perimeter} = \text{Original Perimeter} + 2\pi r
$$

This formula arises because:
- The original edges are extended outward by $ r $, but their lengths remain the same.
- The corners of the polygon are replaced by **circular arcs** of radius $ r $, and the total angle of these arcs is $ 2\pi $, contributing a total arc length of $ 2\pi r $.

---

### Step 3: Apply to the Given Nonagon

- The nonagon is **regular** with **side length 2**.
- It has **9 sides**, so its **original perimeter** is:

$$
P = 9 \times 2 = 18
$$

- The buffer zone is **1 unit wide**, so $ r = 1 $.

Thus, the perimeter of the expanded shape (i.e., the perimeter of $ S $) is:

$$
P_S = 18 + 2\pi
$$

---

### Final Answer

$$
\boxed{18 + 2\pi}
$$

Solution:
$S$ looks like a nonagon with slightly rounded corners.  We draw adjacent sides of the nonagon and look at the boundary of $S$:

[asy]
size(200);
draw((-7.66,-6.43)--(0,0)--(10,0)--(17.66,-6.43));
draw((0,5)--(10,5),blue); draw((13.21,3.83)--(20.87,-2.60),blue);
draw(Arc((10,0),5,50,90),red); draw(Arc((0,0),5,90,130),red);
draw((10,0)--(10,5),dashed); draw((0,0)--(0,5),dashed);
draw((10,0)--(13.21,3.83),dashed);
label("2",(5,0),S); label("1",(10,2.5),W);
draw((-3.21,3.83)--(-10.87,-2.60),blue);
draw((-3.21,3.83)--(0,0),dashed);
[/asy] We can split the portion of $S$ that is outside the nonagon into 9 rectangles and 9 circle sectors, thereby breaking the perimeter of $S$ into alternating straight lines (colored blue above) and curved arcs (colored red above).  The perimeter of $S$ is comprised of nine blue lines and nine red arcs.

Each rectangle has side lengths 1 and 2, so each blue line is 2 units long and the total length of the blue portion of the perimeter is $2\cdot 9 = 18$ units.

Around each vertex of the nonagon, an interior angle, two right angles, and an angle of the circular sector add up to 360 degrees.  The angles inside a nonagon each measure $180(9-2)/9=140$ degrees.  Thus, each circular sector angle measures $360-90-90-140=40$ degrees.  Each sector has radius 1 and arc length $\frac{40^\circ}{360^\circ}(2)(\pi)(1)=\frac{1}{9}(2\pi)$, so nine of these sectors have total arc length $2\pi$.  Thus the total length of the red portion of the perimeter is $2\pi$ units.  (Notice that this is equal to the perimeter of a circle with radius 1, which is what the nine sectors add up to.)

Finally, the perimeter of $S$ is $\boxed{18+2\pi}$ units.


Explanation:
Okay, so I need to figure out for how many real values of x the expression √(120 - √x) is an integer. Let me start by understanding the problem step by step.

First, let me note that the expression inside the square root, which is 120 - √x, must be non-negative because you can't take the square root of a negative number in the real number system. So, that gives me the inequality:

120 - √x ≥ 0
Which simplifies to:
√x ≤ 120
Then squaring both sides (since both sides are non-negative):
x ≤ 120²
So x ≤ 14400.
Also, since √x is defined, x must be non-negative. So x ≥ 0.
Therefore, x is in the interval [0, 14400].

Now, the main condition is that √(120 - √x) is an integer. Let me denote that integer as n. So:

√(120 - √x) = n, where n is an integer.
Then, squaring both sides:
120 - √x = n²
Which rearranges to:
√x = 120 - n²
Then, squaring both sides again:
x = (120 - n²)²

But since x must be real and non-negative, we need to ensure that (120 - n²)² is non-negative. However, since any real number squared is non-negative, this will always hold. However, we also have the earlier condition that √x must be real, which we already considered. But also, from the first step, we had that √x ≤ 120, so 120 - n² must be non-negative because √x is non-negative. Wait, let me check that.

Wait, √x = 120 - n². Since √x is non-negative, then 120 - n² must be ≥ 0. Therefore:

120 - n² ≥ 0
Which implies:
n² ≤ 120
Therefore, n² ≤ 120. Since n is an integer, n can be any integer such that n² ≤ 120. Also, since √(120 - √x) is a square root, it must be non-negative. Therefore, n must be a non-negative integer. So n ≥ 0.

So, n is a non-negative integer with n² ≤ 120. Let me find all such integers n.

First, let's find the maximum possible value of n. Since n² ≤ 120, then n ≤ √120. Let me compute √120. Well, 10² = 100, 11² = 121. So √120 is between 10 and 11. Therefore, the maximum integer n can be is 10, since 10² = 100 ≤ 120, and 11² = 121 > 120. Therefore, n can be 0, 1, 2, ..., 10. So that's 11 possible values of n.

But wait, hold on. Let me check if n can be zero. If n = 0, then √(120 - √x) = 0, which implies 120 - √x = 0, so √x = 120, so x = 120² = 14400. Which is valid. So n=0 is allowed. Similarly, n=10 gives √x = 120 - 10² = 120 - 100 = 20, so x = 20² = 400. Which is also valid. So n ranges from 0 to 10 inclusive. Therefore, there are 11 possible integer values for n. Each of these gives a unique x, right?

Wait, but the question is asking for how many real values of x. So if each n gives exactly one x, then there are 11 real values of x. But wait, let me check if different n's can lead to the same x. Suppose two different integers n1 and n2 give the same x. Then:

(120 - n1²)² = (120 - n2²)²
Which implies that 120 - n1² = ±(120 - n2²).
But since 120 - n² is non-negative (as we had earlier), then 120 - n1² and 120 - n2² are both non-negative. Therefore, the equation becomes:

120 - n1² = 120 - n2²
Which implies n1² = n2². Since n1 and n2 are non-negative integers, this implies n1 = n2. Therefore, each n gives a unique x. Therefore, there are 11 real values of x. Therefore, the answer is 11?

Wait, but let me check again. Let me think if there are any constraints I missed. For example, when I squared both sides of equations, could there be extraneous solutions? Let me check.

Starting with √(120 - √x) = n. Since n is an integer, and square roots are non-negative, n must be non-negative. Then, solving gives √x = 120 - n². Since √x is non-negative, 120 - n² must be non-negative, so n² ≤ 120. So n can be 0, 1, ..., 10. Each of these gives a unique x. Therefore, 11 values. Therefore, the answer is 11?

But wait, let me check with specific examples. For example, take n=0: x=(120 - 0)^2=14400. Then √x=120, so √(120 - √x)=√(120 - 120)=√0=0, which is integer. Good. For n=1: x=(120 - 1)^2=119²=14161. Then √x=119, so √(120 - 119)=√1=1, which is integer. Similarly, n=10: x=(120 - 100)^2=20²=400. Then √x=20, so √(120 - 20)=√100=10, which is integer. So all these check out.

But wait, is there a possibility that for some n, 120 - n² is negative? But we already considered that n² ≤ 120, so 120 - n² is non-negative. Therefore, all these x's are valid. Therefore, 11 values. But wait, hold on. Wait, the problem says "real values of x". But each n gives exactly one x. So 11 different x's. Therefore, the answer is 11?

But wait, let me check if there are other possible n's. For example, could n be negative? But since √(120 - √x) is a square root, it's non-negative, so n must be non-negative. Therefore, n can't be negative. Therefore, n is from 0 to 10 inclusive. So 11 values. Therefore, answer is 11?

But wait, let me check again. Let me think if there's another way to approach this problem. Maybe I made a mistake in assuming that each n gives a unique x. Let me check for n and -n? But n is non-negative, so no. Wait, but suppose someone thought that maybe n could be negative? But since square roots are non-negative, n must be non-negative. Therefore, n is 0,1,...,10. So 11 values. Therefore, answer is 11. But let me check again.

Alternatively, maybe there's a mistake in the step where I said n² ≤ 120. Let me check that again. From √x = 120 - n². Since √x is non-negative, 120 - n² ≥ 0 => n² ≤ 120. Therefore, n can be integers from 0 up to floor(√120). Since √120 ≈ 10.954, floor(√120) is 10. Therefore, n can be 0,1,...,10. So 11 integers. Therefore, 11 values of x. Therefore, answer is 11. But wait, let me check if maybe for some n, 120 - n² is not an integer? Wait, no. Wait, √x is equal to 120 - n². But √x is a real number, but x is real. However, x is (120 - n²)^2, which is real regardless of n. However, the problem says "real values of x", so even if √x is not an integer, x is still real. Wait, but in this case, √x is 120 - n², which is real because n is integer and 120 - n² is real. Therefore, x is real. Therefore, all these 11 values are valid. Therefore, answer is 11?

But wait, hold on. Wait, the problem says "for how many real values of x is √(120 - √x) an integer". So each n gives exactly one x. Therefore, 11 values. But maybe there's something wrong here. Let me check with n=10. Then x=(120 - 10²)^2=(120 - 100)^2=20²=400. So x=400. Then √x=20. Then √(120 - √x)=√(120 - 20)=√100=10, which is integer. Similarly, for n=10, that works. For n=11, if we tried n=11, then √x=120 - 121= -1, which is invalid because √x can't be negative. Therefore, n=11 is invalid. Therefore, n can only go up to 10. Therefore, 11 values. Therefore, answer is 11. But wait, let me check if maybe there are other x's that satisfy the condition. For example, suppose that √(120 - √x) is integer, but maybe there are multiple n's that can lead to same x? Wait, earlier I thought that each n gives unique x. Let me check for example, if there is some overlap. Suppose for two different n's, say n and m, we have (120 - n²)^2 = (120 - m²)^2. Then, as before, 120 - n² = ±(120 - m²). But since 120 - n² and 120 - m² are non-negative, then 120 - n² = 120 - m², so n² = m². Since n and m are non-negative integers, this implies n = m. Therefore, each n gives unique x. Therefore, 11 different x's. Therefore, answer is 11. But wait, let me check with specific numbers. For example, take n=0 and n= something else? For n=0, x=120²=14400. For n=1, x=119²=14161. For n=2, x=118²=13924. And so on, decreasing by 1 each time. So each n gives a unique x. Therefore, 11 different x's. Therefore, answer is 11. But wait, the problem says "real values of x". So even if x is same for different n? But we saw that they are unique. Therefore, answer is 11. But wait, I have a doubt here because sometimes problems like this can have more solutions. Let me think again.

Alternatively, maybe I need to consider that √x must be real, so x ≥ 0, which we already considered. Also, 120 - √x must be non-negative, so √x ≤ 120, so x ≤ 14400. So x is between 0 and 14400. Now, for each integer n ≥ 0, such that n² ≤ 120, we have x=(120 - n²)^2. Therefore, the number of such x is equal to the number of integers n with 0 ≤ n ≤ floor(√120). Since √120 is approx 10.95, floor(√120)=10. Therefore, n=0,...,10, which is 11 numbers. Therefore, 11 real numbers x. Therefore, answer is 11. But maybe the answer is different? Wait, let me check with n=0 to n=10. Let me count them:

n=0,1,2,3,4,5,6,7,8,9,10. That's 11 numbers. So 11 values. Therefore, answer is 11. But maybe I need to check if for some n, (120 - n²) is zero? For example, when n²=120. But n must be integer, so n²=120 would require n=√120, which is not integer. Therefore, there is no integer n with n²=120. Therefore, all the x's are distinct. Therefore, answer is 11. Therefore, I think the answer is 11. But let me check again if there's a mistake here.

Wait, but let me think again. Suppose someone says that √(120 - √x) is integer. Let me denote y = √x. Then, since x is real and non-negative, y is real and non-negative. Then, the equation becomes √(120 - y) is integer. Let me denote that integer as n. Then:

√(120 - y) = n
Which implies 120 - y = n²
Therefore, y = 120 - n²
But since y = √x, then √x = 120 - n²
Therefore, x = (120 - n²)²
Which is the same as before. Also, since y = √x ≥ 0, then 120 - n² ≥ 0 => n² ≤ 120. Therefore, same conclusion. Therefore, n is integer from 0 to 10 inclusive. Therefore, 11 values. Therefore, answer is 11. Therefore, I think that is correct.

But maybe the problem is in the interpretation of "real values of x". For example, if different n's could lead to same x? But as we saw, since x = (120 - n²)^2, and n is unique for each x, because if (120 - n²)^2 = (120 - m²)^2, then 120 - n² = ±(120 - m²). But since 120 - n² and 120 - m² are non-negative, then 120 - n² = 120 - m², so n² = m², so n = m. Therefore, each x corresponds to exactly one n. Therefore, 11 different x's. Therefore, answer is 11. Therefore, I think the answer is 11. But maybe I need to check if there are other constraints?

Wait, but let me check for n=0, x=(120 - 0)^2=14400. Then √x=120, so √(120 - √x)=√(120 - 120)=√0=0, which is integer. For n=1, x=(120 - 1)^2=119²=14161. Then √x=119, so √(120 - 119)=√1=1, integer. Similarly, n=2: x=(120 - 4)=116, so √x=116? Wait, no. Wait, x=(120 - n²)^2. So for n=2, x=(120 - 4)^2=116²=13456. Then √x=116, so √(120 - 116)=√4=2, which is integer. So that works. Similarly, for n=10, x=(120 - 100)^2=20²=400. Then √x=20, so √(120 - 20)=√100=10, integer. So all of them work. Therefore, 11 values. Therefore, answer is 11. Therefore, I think that is correct.

But wait, let me check if there are other possible values. For example, suppose that √(120 - √x) is integer, but maybe there are other solutions where √x is not equal to 120 - n²? Wait, no. Because if √(120 - √x) is integer, then by definition, we set that equal to n, and then solved for √x. Therefore, all solutions must satisfy √x = 120 - n². Therefore, there are no other solutions. Therefore, the total number of real x is equal to the number of integers n with n² ≤ 120, which is 11. Therefore, answer is 11. Therefore, I think that is correct.

But wait, hold on. Let me check if n can be zero. If n=0, then √(120 - √x)=0, which implies 120 - √x=0, so √x=120, so x=14400. Which is valid. So that's one value. Similarly, n=1 gives another value, etc. So 11 values. Therefore, answer is 11. Therefore, I think that is correct.

But maybe the answer is different? Wait, let me check with n=10. If n=10, then √x=120 - 100=20, so x=400. If n= something else? For example, if n= sqrt(120 - sqrt(x)), but I think we already considered all possibilities. Therefore, I think the answer is 11. Therefore, I think the answer is \boxed{11}.

Wait, but hold on. Wait, let me check if there are any restrictions on x. For example, x must be such that √x is real, which we already considered. Also, 120 - √x must be non-negative, which we also considered. Therefore, all the x's we found are valid. Therefore, 11 values. Therefore, answer is 11. Therefore, I think that is correct.

But maybe I need to check if there are other integers n where n² ≤ 120. Let me list them:

n=0: 0²=0 ≤120
n=1:1
n=2:4
n=3:9
n=4:16
n=5:25
n=6:36
n=7:49
n=8:64
n=9:81
n=10:100
n=11:121>120. So yes, n=0,...,10. So 11 values. Therefore, answer is 11. Therefore, I think that is correct.

But maybe the problem is in the fact that x must be real. But since x is defined as (120 - n²)^2, which is real for any real number, and since n is integer, x is real. Therefore, all 11 values are real. Therefore, answer is 11. Therefore, I think that is correct.

But let me check once again. Suppose someone says that maybe for some n, 120 - n² is not an integer? But no, because n is integer, so 120 - n² is integer. Therefore, √x is integer? Wait, no. Wait, √x is equal to 120 - n², which is integer. Therefore, √x is integer. Therefore, x is a perfect square. Therefore, x is a perfect square. Therefore, for each n, x is a perfect square. Therefore, there are 11 such x's. Therefore, answer is 11. Therefore, I think that is correct.

Therefore, I think the answer is 11. Therefore, I will go with \boxed{11}.

**Final Answer**
\boxed{11}

To determine how many real values of $ x $ make the expression $ \sqrt{120 - \sqrt{x}} $ an integer, we proceed step by step.

---

### Step 1: Understand the Expression

We are given the expression:

$$
\sqrt{120 - \sqrt{x}}
$$

We are told this expression must be an **integer**. Let’s denote this integer by $ n $, so:

$$
\sqrt{120 - \sqrt{x}} = n
$$

Since square roots are non-negative, $ n \geq 0 $, and $ n $ must be an integer.

---

### Step 2: Solve for $ x $

Squaring both sides:

$$
120 - \sqrt{x} = n^2
$$

Solving for $ \sqrt{x} $:

$$
\sqrt{x} = 120 - n^2
$$

Now square both sides again to solve for $ x $:

$$
x = (120 - n^2)^2
$$

This gives us a formula for $ x $ in terms of $ n $.

---

### Step 3: Determine Valid Values of $ n $

We must ensure that $ \sqrt{x} = 120 - n^2 $ is **non-negative**, because square roots are defined only for non-negative numbers. So:

$$
120 - n^2 \geq 0 \Rightarrow n^2 \leq 120
$$

We now find all **non-negative integers** $ n $ such that $ n^2 \leq 120 $.

Let’s compute the square of integers:

- $ 0^2 = 0 $
- $ 1^2 = 1 $
- $ 2^2 = 4 $
- $ 3^2 = 9 $
- $ 4^2 = 16 $
- $ 5^2 = 25 $
- $ 6^2 = 36 $
- $ 7^2 = 49 $
- $ 8^2 = 64 $
- $ 9^2 = 81 $
- $ 10^2 = 100 $
- $ 11^2 = 121 $ (too large)

So the valid values of $ n $ are:

$$
n = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
$$

That’s a total of **11** values.

---

### Step 4: Verify Uniqueness of $ x $

Each value of $ n $ gives a unique value of $ x $, since:

$$
x = (120 - n^2)^2
$$

If two different values of $ n $, say $ n_1 $ and $ n_2 $, gave the same $ x $, then:

$$
(120 - n_1^2)^2 = (120 - n_2^2)^2
$$

Taking square roots (and noting both sides are non-negative):

$$
120 - n_1^2 = 120 - n_2^2 \Rightarrow n_1^2 = n_2^2
$$

Since $ n_1, n_2 \geq 0 $, this implies $ n_1 = n_2 $. Therefore, each $ n $ gives a **unique** $ x $.

---

### Step 5: Final Answer

There are exactly **11** non-negative integers $ n $ such that $ n^2 \leq 120 $, and each gives a unique real value of $ x $. Therefore, the number of real values of $ x $ for which $ \sqrt{120 - \sqrt{x}} $ is an integer is:

$$
\boxed{11}
$$

[Assistant]
Explanation:
Okay, so I need to figure out for how many real values of x the expression √(120 - √x) is an integer. Let me start by understanding the problem step by step.

First, let me note that the expression inside the square root, which is 120 - √x, must be non-negative because you can't take the square root of a negative number in the real number system. So, that gives me the inequality:

120 - √x ≥ 0
Which simplifies to:
√x ≤ 120
Then squaring both sides (since both sides are non-negative):
x ≤ 120²
So x ≤ 14400.
Also, since √x is defined, x must be non-negative. So x ≥ 0.
Therefore, x is in the interval [0, 14400].

Now, the main condition is that √(120 - √x) is an integer. Let me denote that integer as n. So:

√(120 - √x) = n, where n is an integer.
Then, squaring both sides:
120 - √x = n²
Which rearranges to:
√x = 120 - n²
Then, squaring both sides again:
x = (120 - n²)²

But since x must be real and non-negative, we need to ensure that (120 - n²)² is non-negative. However, since any real number squared is non-negative, this will always hold. However, we also have the earlier condition that √x must be real, which we already considered. But also, from the first step, we had that √x ≤ 120, so 120 - n² must be non-negative because √x is non-negative. Wait, let me check that.

Wait, √x = 120 - n². Since √x is non-negative, then 120 - n² must be ≥ 0. Therefore:

120 - n² ≥ 0
Which implies:
n² ≤ 120
Therefore, n² ≤ 120. Since n is an integer, n can be any integer such that n² ≤ 120. Also, since √(120 - √x) is a square root, it must be non-negative. Therefore, n must be a non-negative integer. So n ≥ 0.

So, n is a non-negative integer with n² ≤ 120. Let me find all such integers n.

First, let's find the maximum possible value of n. Since n² ≤ 120, then n ≤ √120. Let me compute √120. Well, 10² = 100, 11² = 121. So √120 is between 10 and 11. Therefore, the maximum integer n can be is 10, since 10² = 100 ≤ 120, and 11² = 121 > 120. Therefore, n can be 0, 1, 2, ..., 10. So that's 11 possible values of n.

But wait, hold on. Let me check if n can be zero. If n = 0, then √(120 - √x) = 0, which implies 120 - √x = 0, so √x = 120, so x = 120² = 14400. Which is valid. So n=0 is allowed. Similarly, n=10 gives √x = 120 - 10² = 120 - 100 = 20, so x = 20² = 400. Which is also valid. So n ranges from 0 to 10 inclusive. Therefore, there are 11 possible integer values for n. Each of these gives a unique x, right?

Wait, but the question is asking for how many real values of x. So if each n gives exactly one x, then there are 11 real values of x. But wait, let me check if different n's can lead to the same x. Suppose two different integers n1 and n2 give the same x. Then:

(120 - n1²)² = (120 - n2²)²
Which implies that 120 - n1² = ±(120 - n2²).
But since 120 - n² is non-negative (as we had earlier), then 120 - n1² and 120 - n2² are both non-negative. Therefore, the equation becomes:

120 - n1² = 120 - n2²
Which implies n1² = n2². Since n1 and n2 are non-negative integers, this implies n1 = n2. Therefore, each n gives a unique x. Therefore, there are 11 real values of x. Therefore, the answer is 11?

Wait, but let me check again. Let me think if there are any constraints I missed. For example, when I squared both sides of equations, could there be extraneous solutions? Let me check.

Starting with √(120 - √x) = n. Since n is an integer, and square roots are non-negative, n must be non-negative. Then, solving gives √x = 120 - n². Since √x is non-negative, 120 - n² must be non-negative, so n² ≤ 120. So n can be 0, 1, ..., 10. Each of these gives a unique x. Therefore, 11 values. Therefore, the answer is 11?

But wait, let me check with specific examples. For example, take n=0: x=(120 - 0)^2=14400. Then √x=120, so √(120 - √x)=√(120 - 120)=√0=0, which is integer. Good. For n=1: x=(120 - 1)^2=119²=14161. Then √x=119, so √(120 - 119)=√1=1, which is integer. Similarly, n=10: x=(120 - 100)^2=20²=400. Then √x=20, so √(120 - 20)=√100=10, which is integer. So all these check out.

But wait, is there a possibility that for some n, 120 - n² is negative? But we already considered that n² ≤ 120, so 120 - n² is non-negative. Therefore, all these x's are valid. Therefore, 11 values. But wait, hold on. Wait, the problem says "real values of x". But each n gives exactly one x. So 11 different x's. Therefore, the answer is 11?

But wait, let me check if there are other possible n's. For example, could n be negative? But since √(120 - √x) is a square root, it's non-negative, so n must be non-negative. Therefore, n can't be negative. Therefore, n is from 0 to 10 inclusive. So 11 values. Therefore, answer is 11?

But wait, let me check again. Let me think if there's another way to approach this problem. Maybe I made a mistake in assuming that each n gives a unique x. Let me check for n and -n? But n is non-negative, so no. Wait, but suppose someone thought that maybe n could be negative? But since square roots are non-negative, n must be non-negative. Therefore, n is 0,1,...,10. So 11 values. Therefore, answer is 11. But let me check again.

Alternatively, maybe there's a mistake in the step where I said n² ≤ 120. Let me check that again. From √x = 120 - n². Since √x is non-negative, 120 - n² ≥ 0 => n² ≤ 120. Therefore, n can be integers from 0 up to floor(√120). Since √120 ≈ 10.954, floor(√120) is 10. Therefore, n can be 0,1,...,10. So 11 integers. Therefore, 11 values of x. Therefore, answer is 11. But wait, let me check if maybe for some n, 120 - n² is not an integer? Wait, no. Wait, √x is equal to 120 - n². But √x is a real number, but x is real. However, x is (120 - n²)^2, which is real regardless of n. However, the problem says "real values of x", so even if √x is not an integer, x is still real. Wait, but in this case, √x is 120 - n², which is real because n is integer and 120 - n² is real. Therefore, x is real. Therefore, all these 11 values are valid. Therefore, answer is 11?

But wait, hold on. Wait, the problem says "for how many real values of x is √(120 - √x) an integer". So each n gives exactly one x. Therefore, 11 values. But maybe there's something wrong here. Let me check with n=10. Then x=(120 - 10²)^2=(120 - 100)^2=20²=400. So x=400. Then √x=20. Then √(120 - √x)=√(120 - 20)=√100=10, which is integer. Similarly, for n=10, that works. For n=11, if we tried n=11, then √x=120 - 121= -1, which is invalid because √x can't be negative. Therefore, n=11 is invalid. Therefore, n can only go up to 10. Therefore, 11 values. Therefore, answer is 11. But wait, let me check if maybe there are other x's that satisfy the condition. For example, suppose that √(120 - √x) is integer, but maybe there are multiple n's that can lead to same x? Wait, earlier I thought that each n gives unique x. Let me check for example, if there is some overlap. Suppose for two different n's, say n and m, we have (120 - n²)^2 = (120 - m²)^2. Then, as before, 120 - n² = ±(120 - m²). But since 120 - n² and 120 - m² are non-negative, then 120 - n² = 120 - m², so n² = m². Since n and m are non-negative integers, this implies n = m. Therefore, each n gives unique x. Therefore, 11 different x's. Therefore, answer is 11. But wait, let me check with specific numbers. For example, take n=0 and n= something else? For n=0, x=120²=14400. For n=1, x=119²=14161. For n=2, x=118²=13924. And so on, decreasing by 1 each time. So each n gives a unique x. Therefore, 11 different x's. Therefore, answer is 11. But wait, the problem says "real values of x". So even if x is same for different n? But we saw that they are unique. Therefore, answer is 11. But wait, I have a doubt here because sometimes problems like this can have more solutions. Let me think again.

Alternatively, maybe I need to consider that √x must be real, so x ≥ 0, which we already considered. Also, 120 - √x must be non-negative, so √x ≤ 120, so x ≤ 14400. So x is between 0 and 14400. Now, for each integer n ≥ 0, such that n² ≤ 120, we have x=(120 - n²)^2. Therefore, the number of such x is equal to the number of integers n with 0 ≤ n ≤ floor(√120). Since √120 is approx 10.95, floor(√120)=10. Therefore, n=0,...,10, which is 11 numbers. Therefore, 11 real numbers x. Therefore, answer is 11. But maybe the answer is different? Wait, let me check with n=0 to n=10. Let me count them:

n=0,1,2,3,4,5,6,7,8,9,10. That's 11 numbers. So 11 values. Therefore, answer is 11. But maybe I need to check if for some n, (120 - n²) is zero? For example, when n²=120. But n must be integer, so n²=120 would require n=√120, which is not integer. Therefore, there is no integer n with n²=120. Therefore, all the x's are distinct. Therefore, answer is 11. Therefore, I think the answer is 11. But let me check again if there's a mistake here.

Wait, but let me think again. Suppose someone says that √(120 - √x) is integer. Let me denote y = √x. Then, since x is real and non-negative, y is real and non-negative. Then, the equation becomes √(120 - y) is integer. Let me denote that integer as n. Then:

√(120 - y) = n
Which implies 120 - y = n²
Therefore, y = 120 - n²
But since y = √x, then √x = 120 - n²
Therefore, x = (120 - n²)²
Which is the same as before. Also, since y = √x ≥ 0, then 120 - n² ≥ 0 => n² ≤ 120. Therefore, same conclusion. Therefore, n is integer from 0 to 10 inclusive. Therefore, 11 values. Therefore, answer is 11. Therefore, I think that is correct.

But maybe the problem is in the interpretation of "real values of x". For example, if different n's could lead to same x? But as we saw, since x = (120 - n²)^2, and n is unique for each x, because if (120 - n²)^2 = (120 - m²)^2, then 120 - n² = ±(120 - m²). But since 120 - n² and 120 - m² are non-negative, then 120 - n² = 120 - m², so n² = m², so n = m. Therefore, each x corresponds to exactly one n. Therefore, 11 different x's. Therefore, answer is 11. Therefore, I think the answer is 11. But maybe I need to check if there are other constraints?

Wait, but let me check for n=0, x=(120 - 0)^2=14400. Then √x=120, so √(120 - √x)=√(120 - 120)=√0=0, which is integer. For n=1, x=(120 - 1)^2=119²=14161. Then √x=119, so √(120 - 119)=√1=1, integer. Similarly, n=2: x=(120 - 4)=116, so √x=116? Wait, no. Wait, x=(120 - n²)^2. So for n=2, x=(120 - 4)^2=116²=13456. Then √x=116, so √(120 - 116)=√4=2, which is integer. So that works. Similarly, for n=10, x=(120 - 100)^2=20²=400. Then √x=20, so √(120 - 20)=√100=10, integer. So all of them work. Therefore, 11 values. Therefore, answer is 11. Therefore, I think that is correct.

But wait, let me check if there are other possible values. For example, suppose that √(120 - √x) is integer, but maybe there are other solutions where √x is not equal to 120 - n²? Wait, no. Because if √(120 - √x) is integer, then by definition, we set that equal to n, and then solved for √x. Therefore, all solutions must satisfy √x = 120 - n². Therefore, there are no other solutions. Therefore, the total number of real x is equal to the number of integers n with n² ≤ 120, which is 11. Therefore, answer is 11. Therefore, I think that is correct.

But wait, hold on. Let me check if n can be zero. If n=0, then √(120 - √x)=0, which implies 120 - √x=0, so √x=120, so x=14400. Which is valid. So that's one value. Similarly, n=1 gives another value, etc. So 11 values. Therefore, answer is 11. Therefore, I think that is correct.

But maybe the answer is different? Wait, let me check with n=10. If n=10, then √x=120 - 100=20, so x=400. If n= something else? For example, if n= sqrt(120 - sqrt(x)), but I think we already considered all possibilities. Therefore, I think the answer is 11. Therefore, I think the answer is \boxed{11}.

Wait, but hold on. Wait, let me check if there are any restrictions on x. For example, x must be such that √x is real, which we already considered. Also, 120 - √x must be non-negative, which we also considered. Therefore, all the x's we found are valid. Therefore, 11 values. Therefore, answer is 11. Therefore, I think that is correct.

But maybe I need to check if there are other integers n where n² ≤ 120. Let me list them:

n=0: 0²=0 ≤120
n=1:1
n=2:4
n=3:9
n=4:16
n=5:25
n=6:36
n=7:49
n=8:64
n=9:81
n=10:100
n=11:121>120. So yes, n=0,...,10. So 11 values. Therefore, answer is 11. Therefore, I think that is correct.

But maybe the problem is in the fact that x must be real. But since x is defined as (120 - n²)^2, which is real for any real number, and since n is integer, x is real. Therefore, all 11 values are real. Therefore, answer is 11. Therefore, I think that is correct.

But let me check once again. Suppose someone says that maybe for some n, 120 - n² is not an integer? But no, because n is integer, so 120 - n² is integer. Therefore, √x is integer? Wait, no. Wait, √x is equal to 120 - n², which is integer. Therefore, √x is integer. Therefore, x is a perfect square. Therefore, x is a perfect square. Therefore, for each n, x is a perfect square. Therefore, there are 11 such x's. Therefore, answer is 11. Therefore, I think that is correct.

Therefore, I think the answer is 11. Therefore, I will go with \boxed{11}.

**Final Answer**
\boxed{11}

To determine how many real values of $ x $ make the expression $ \sqrt{120 - \sqrt{x}} $ an integer, we proceed step by step.

---

### Step 1: Understand the Expression

We are given the expression:

$$
\sqrt{120 - \sqrt{x}}
$$

We are told this expression must be an **integer**. Let’s denote this integer by $ n $, so:

$$
\sqrt{120 - \sqrt{x}} = n
$$

Since square roots are non-negative, $ n \geq 0 $, and $ n $ must be an integer.

---

### Step 2: Solve for $ x $

Squaring both sides:

$$
120 - \sqrt{x} = n^2
$$

Solving for $ \sqrt{x} $:

$$
\sqrt{x} = 120 - n^2
$$

Now square both sides again to solve for $ x $:

$$
x = (120 - n^2)^2
$$

This gives us a formula for $ x $ in terms of $ n $.

---

### Step 3: Determine Valid Values of $ n $

We must ensure that $ \sqrt{x} = 120 - n^2 $ is **non-negative**, because square roots are defined only for non-negative numbers. So:

$$
120 - n^2 \geq 0 \Rightarrow n^2 \leq 120
$$

We now find all **non-negative integers** $ n $ such that $ n^2 \leq 120 $.

Let’s compute the square of integers:

- $ 0^2 = 0 $
- $ 1^2 = 1 $
- $ 2^2 = 4 $
- $ 3^2 = 9 $
- $ 4^2 = 16 $
- $ 5^2 = 25 $
- $ 6^2 = 36 $
- $ 7^2 = 49 $
- $ 8^2 = 64 $
- $ 9^2 = 81 $
- $ 10^2 = 100 $
- $ 11^2 = 121 $ (too large)

So the valid values of $ n $ are:

$$
n = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
$$

That’s a total of **11** values.

---

### Step 4: Verify Uniqueness of $ x $

Each value of $ n $ gives a unique value of $ x $, since:

$$
x = (120 - n^2)^2
$$

If two different values of $ n $, say $ n_1 $ and $ n_2 $, gave the same $ x $, then:

$$
(120 - n_1^2)^2 = (120 - n_2^2)^2
$$

Taking square roots (and noting both sides are non-negative):

$$
120 - n_1^2 = 120 - n_2^2 \Rightarrow n_1^2 = n_2^2
$$

Since $ n_1, n_2 \geq 0 $, this implies $ n_1 = n_2 $. Therefore, each $ n $ gives a **unique** $ x $.

---

### Step 5: Final Answer

There are exactly **11** non-negative integers $ n $ such that $ n^2 \leq 120 $, and each gives a unique real value of $ x $. Therefore, the number of real values of $ x $ for which $ \sqrt{120 - \sqrt{x}} $ is an integer is:

$$
\boxed{11}
$$

Solution:
Suppose that  $k = \sqrt{120 - \sqrt{x}}$ is an integer. Then $0\le k \le \sqrt{120}$, and because $k$ is an integer, we have $0\le k \le 10$. Thus there are 11 possible integer values of $k$. For each such $k$, the corresponding value of $x$ is $\left(120 - k^2\right)^2$. Because  $\left(120 -
k^2\right)^2$ is positive and decreasing for $0\le k \le 10$, the $\boxed{11}$ values of $x$ are distinct.