Simon has built a gazebo, whose shape is a regular heptagon, with a side length of $3$ units. He has also built a walkway around the gazebo, of constant width $2$ units, as shown below. (Every point on the ground that is within $2$ units of the gazebo and outside the gazebo is covered by the walkway.) Find the area of the walkway.

Asymptote code below
[asy]
unitsize(1 cm);

pair A, B;

for(int i = 0; i <= 6; ++i) {
A = dir(90 + 360/7*i);
B = dir(90 + 360/7*(i + 1));
fill(A--arc(A, 1/2*abs(A - B), 90 + 360/7*i - 30, 90 + 360/7*i + 30)--cycle, gray(0.8));
fill(A--(A + scale(1/2)*rotate(90)*(A - B))--(B + scale(1/2)*rotate(90)*(A - B))--B--cycle, gray(0.8));
}

for(int i = 0; i <= 6; ++i) {
A = dir(90 + 360/7*i);
B = dir(90 + 360/7*(i + 1));
draw(A--B);
draw(arc(A, 1/2*abs(A - B), 90 + 360/7*i - 360/7/2, 90 + 360/7*i + 360/7/2));
draw((A + scale(1/2)*rotate(90)*(A - B))--(B + scale(1/2)*rotate(90)*(A - B)));
}

[/asy]


To solve this problem, we need to calculate the area of the walkway surrounding the heptagonal gazebo. The walkway extends 2 units around the gazebo, which is shaped like a regular heptagon with side length 3 units. Our task is to find the area of the walkway.

### Step 1: Area of the heptagon
The area $ A_{\text{heptagon}} $ of a regular polygon can be calculated using the formula:
\[
A_{\text{polygon}} = \frac{1}{4}n s^2 \cot\left(\frac{\pi}{n}\right)
\]
where:
- $ n $ is the number of sides,
- $ s $ is the side length of the polygon.

For a heptagon, $ n = 7 $ and the side length $ s = 3 $. Plugging these values into the formula:

\[
A_{\text{heptagon}} = \frac{1}{4} \times 7 \times 3^2 \times \cot\left(\frac{\pi}{7}\right)
\]
Simplifying:
\[
A_{\text{heptagon}} = \frac{1}{4} \times 7 \times 9 \times \cot\left(\frac{\pi}{7}\right)
\]
\[
A_{\text{heptagon}} = \frac{63}{4} \times \cot\left(\frac{\pi}{7}\right)
\]

Using a calculator to find $ \cot\left(\frac{\pi}{7}\right) \approx 0.792 $, we get:
\[
A_{\text{heptagon}} \approx \frac{63}{4} \times 0.792 = 12.474
\]

So, the area of the gazebo is approximately $ 12.474 $ square units.

### Step 2: Area of the larger heptagon
The walkway extends 2 units around the gazebo. This means that the new heptagon (enclosing both the gazebo and walkway) has a side length of $ 3 + 2 = 5 $ units.

Using the same formula to calculate the area of the larger heptagon with side length $ s = 5 $:
\[
A_{\text{large heptagon}} = \frac{1}{4} \times 7 \times 5^2 \times \cot\left(\frac{\pi}{7}\right)
\]
Simplifying:
\[
A_{\text{large heptagon}} = \frac{1}{4} \times 7 \times 25 \times \cot\left(\frac{\pi}{7}\right)
\]
\[
A_{\text{large heptagon}} = \frac{175}{4} \times \cot\left(\frac{\pi}{7}\right)
\]
Using $ \cot\left(\frac{\pi}{7}\right) \approx 0.792 $ again, we get:
\[
A_{\text{large heptagon}} \approx \frac{175}{4} \times 0.792 = 34.65
\]

So, the area of the larger heptagon is approximately $ 34.65 $ square units.

### Step 3: Area of the walkway
The area of the walkway is the difference between the area of the larger heptagon and the area of the gazebo:
\[
A_{\text{walkway}} = A_{\text{large heptagon}} - A_{\text{heptagon}}
\]
\[
A_{\text{walkway}} \approx 34.65 - 12.474 = 22.176
\]

Thus, the area of the walkway is approximately $ \boxed{22.176} $ square units.

### Follow-up questions:
1. How would the solution change if the side length of the gazebo was 4 units instead of 3?
2. Can you find the perimeter of the heptagonal gazebo?
3. How does the area formula for regular polygons relate to the radius of the circumscribed circle?
4. What is the relationship between the number of sides and the area for polygons with the same side length?
5. How would the area of the walkway change if the width of the walkway was 3 units instead of 2?

### Tip:
When dealing with regular polygons, knowing the relationship between the side length, number of sides, and trigonometric functions can greatly simplify area calculations.

Simon has built a gazebo, whose shape is a regular heptagon, with a side length of 3 units. He has also built a walkway around the gazebo, of constant width 2 units, as shown below. (Every point on the ground that is within 2 units of the gazebo and outside the gazebo is covered by the walkway.) Find the area of the walkway.

This problem explores how to calculate the area of a walkway around a regular heptagon-shaped gazebo with a side length of 3 units. Using geometric and trigonometric formulas, learn how to determine the area difference between two polygons. Suitable for students in Grades 10-12.

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