find the area of the regular polygon. give the answer to the nearest tenth. a hexagon with a radius of 12.5ft
a. 259.8ft^2
b. 65.0ft^2
c. 129.9ft^2
d. 406.1ft^2
The formula for the area of a regular polygon is A = (1/2) * n * s * r, where n is the number of sides, s is the length of each side, and r is the radius of the circumscribed circle.
In this case, the hexagon has 6 sides and a radius of 12.5ft. Since a hexagon is composed of 6 equilateral triangles, we can calculate the side length s using the formula for the area of an equilateral triangle:
Area = (sqrt(3)/4) * s^2
A hexagons's area = 6 * Area of an equilateral triangle
Therefore: (sqrt(3)/4) * s^2 = 6 * (1/2) * s * 12.5
(sqrt(3)/4) * s^2 = 37.5 * s
sqrt(3) * s = 150
s = 150 / sqrt(3)
s ≈ 86.6ft
Now that we have the side length s, we can calculate the area of the hexagon:
A = (1/2) * 6 * 86.6 * 12.5
A = 3897.5ft^2
Rounded to the nearest tenth, the area of the regular hexagon is approximately 3897.5ft^2. However, none of the answer choices provided match this calculation, so it's unclear if the options were incomplete or if there is a calculation error.