What is the area of a regular 15-gon with a perimeter of 90 m? (1 point) Responses 528.2 m² 635.1 m² 1,270.3 m² 142,903.1 m²
To find the area of a regular polygon, you can use the formula:
Area = (Perimeter x apothem) / 2
First, calculate the apothem of the 15-gon.
The apothem can be found using the formula:
s = 2 * R * sin(180/15)
s = 2 * R * sin(12)
where R is the radius of the circumcircle of the polygon.
Given that the perimeter of the 15-gon is 90m and there are 15 equal sides, each side is 90/15 = 6m.
The radius of the circumcircle is the distance from the center of the polygon to one of the vertices. This forms a right triangle with the apothem and half of one of the sides. The length of the radius can be found using Pythagoras' theorem:
(6/2)^2 + apothem^2 = R^2
3^2 + apothem^2 = R^2
9 + apothem^2 = R^2
Substitute this back into the apothem formula:
s = 2 * sqrt(R^2) * sin(12)
Calculate the apothem, and then substitute into the area formula to find the area of the 15-gon.
Therefore, the correct answer is 635.1 m².