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. (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.

ABJelly Aug 2, 2024

#1**0 **

To determine the area of the walkway around the regular heptagon-shaped gazebo, we first need to calculate the area of the heptagon and the area of the larger heptagon formed by extending the sides to include the walkway.

### Step 1: Area of the Regular Heptagon (Gazebo)

A regular heptagon has 7 sides, each of length 3 units. The formula for the area of a regular polygon with \( n \) sides, each of length \( s \), is:

\[

\text{Area} = \frac{1}{4} n s^2 \cot \left(\frac{\pi}{n}\right)

\]

For a heptagon (\( n = 7 \)) with side length \( s = 3 \):

\[

\text{Area of the gazebo} = \frac{1}{4} \times 7 \times 3^2 \cot \left(\frac{\pi}{7}\right)

\]

\[

\text{Area of the gazebo} = \frac{1}{4} \times 7 \times 9 \cot \left(\frac{\pi}{7}\right)

\]

\[

\text{Area of the gazebo} = \frac{63}{4} \cot \left(\frac{\pi}{7}\right)

\]

### Step 2: Area of the Larger Heptagon Including the Walkway

The walkway extends 2 units beyond each side of the original heptagon. Therefore, the new side length of the larger heptagon is \( s + 2 \times 2 = s + 4 \). So the new side length is:

\[

s' = 3 + 4 = 7

\]

Now, calculate the area of the larger heptagon with side length 7 units:

\[

\text{Area of the larger heptagon} = \frac{1}{4} \times 7 \times 7^2 \cot \left(\frac{\pi}{7}\right)

\]

\[

\text{Area of the larger heptagon} = \frac{1}{4} \times 7 \times 49 \cot \left(\frac{\pi}{7}\right)

\]

\[

\text{Area of the larger heptagon} = \frac{343}{4} \cot \left(\frac{\pi}{7}\right)

\]

### 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 original heptagon:

\[

\text{Area of the walkway} = \text{Area of the larger heptagon} - \text{Area of the gazebo}

\]

\[

\text{Area of the walkway} = \frac{343}{4} \cot \left(\frac{\pi}{7}\right) - \frac{63}{4} \cot \left(\frac{\pi}{7}\right)

\]

\[

\text{Area of the walkway} = \frac{343 - 63}{4} \cot \left(\frac{\pi}{7}\right)

\]

\[

\text{Area of the walkway} = \frac{280}{4} \cot \left(\frac{\pi}{7}\right)

\]

\[

\text{Area of the walkway} = 70 \cot \left(\frac{\pi}{7}\right)

\]

Hence, the area of the walkway around the gazebo is:

\[

\boxed{70 \cot \left(\frac{\pi}{7}\right)}

\]

bader Aug 2, 2024