Geometry

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:

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