What is the area of a regular 15-gon with a perimeter of 90 m?
(1 point)
A. 528.2m ^ 2
B.635.1m ^ 2
C. 1.270.3m ^ 2
D.142.903.1 m ^ 2
To find the area of a regular polygon, we can use the formula:
Area = (perimeter * apothem) / 2
Since the perimeter of the 15-gon is given as 90 m, the length of each side is 90/15 = 6 m.
To find the apothem, we can use the formula:
Apothem = side length / (2 * tan(180° / n))
where n is the number of sides (in this case, n=15).
Plugging in the values, we get:
Apothem = 6 / (2 * tan(180° / 15))
Apothem ≈ 6 / (2 * tan(12°))
Using a calculator, we find that tan(12°) ≈ 0.21255656167, so:
Apothem ≈ 6 / (2 * 0.21255656167)
Apothem ≈ 6 / 0.42511312334
Apothem ≈ 14.115 m
Now we can calculate the area:
Area = (90 * 14.115) / 2
Area ≈ 635.1 m²
Therefore, the area of the regular 15-gon with a perimeter of 90 m is approximately 635.1 m².
The correct answer is B. 635.1m ^ 2.