Answer :
The perimeter is 4508.305 sq in.
To find the perimeter and area of the regular nonagon (a polygon with nine sides) with the given apothem a = 37.1 inches and side length s = 27.1 inches, we'll follow these steps:
1. Calculate the perimeter (P):
The perimeter of a regular polygon is simply the sum of all its side lengths. Since a nonagon has nine sides, we multiply the side length by 9.
[tex]\[ P = 9s = 9 \times 27.1 \][/tex]
2. Substitute the value of s and calculate:
[tex]\[ P = 9 \times 27.1 = 243.9 \, \text{inches} \][/tex]
So, the perimeter of the nonagon is 243.9 inches.
3. Calculate the area (A):
The area of a regular polygon can be found using the formula:
[tex]\[ A = \frac{1}{2} \times \text{apothem} \times \text{perimeter} \][/tex]
4. Substitute the values of the apothem and perimeter and calculate:
[tex]\[ A = \frac{1}{2} \times 37.1 \times 243.9 \][/tex]
5. Simplify and calculate:
[tex]\[ A = \frac{1}{2} \times 37.1 \times 243.9 = 4508.305 \, \text{square inches} \][/tex]
So, the area of the nonagon is 4508.305 square inches.
