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Find the perimeter and the area of the regular nonagon with the given apothem [tex]a = 37.1 \, \text{in.}[/tex] and side length [tex]s = 27.1 \, \text{in.}[/tex].

The perimeter is _____ in. (Round to three decimal places as needed.)

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.

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