What is the area of a regular 15-gon with a perimeter of 90 m?
(1 point)
Responses
528.2 m²
528.2 m²
635.1 m²
635.1 m²
1,270.3 m²
1,270.3 m²
142,903.1 m²
142,903.1 m²
To find the area of a regular 15-gon, we need to use the formula:
Area = (perimeter * apothem) / 2
The apothem is the distance from the center of the polygon to the midpoint of any side. In a regular 15-gon, the apothem can be found using the formula:
apothem = s / (2 * tan(180/15))
Where s is the length of any side.
Given that the perimeter is 90 m, we can find the length of each side by dividing the perimeter by 15:
s = 90 / 15 = 6 m
Now we can substitute the value of s into the apothem formula:
apothem = 6 / (2 * tan(180/15))
Using a calculator, we can find that tan(180/15) ≈ 0.2679.
apothem = 6 / (2 * 0.2679) ≈ 11.21 m
Finally, we can substitute the values of the perimeter and the apothem into the area formula:
Area = (90 * 11.21) / 2 ≈ 502.95 m²
Therefore, the area of the regular 15-gon with a perimeter of 90 m is approximately 502.95 m².