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.
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:
Area=14ns2cot(πn)
For a heptagon (n=7) with side length s=3:
Area of the gazebo=14×7×32cot(π7)
Area of the gazebo=14×7×9cot(π7)
Area of the gazebo=634cot(π7)
### 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+2×2=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:
Area of the larger heptagon=14×7×72cot(π7)
Area of the larger heptagon=14×7×49cot(π7)
Area of the larger heptagon=3434cot(π7)
### 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:
Area of the walkway=Area of the larger heptagon−Area of the gazebo
Area of the walkway=3434cot(π7)−634cot(π7)
Area of the walkway=343−634cot(π7)
Area of the walkway=2804cot(π7)
Area of the walkway=70cot(π7)
Hence, the area of the walkway around the gazebo is:
70cot(π7)