Given:
shape: regular octagon (8 sided polygon)
side length : 8 feet
apothem 9.66 feet.
Area of a regular octagon = 2(1+ √2) a²
Area of a regular octagon = 2(1 + 1.414) * (8ft)²
A = 2(2.414)* 64 ft²
A = 4.828 * 64 ft²
A = 308.992 ft² or 309 ft² Choice D.
Since the apothem is given, we can use it to solve for the area of the regular octagon. An octagon has 8 sides, it forms 8 congruent isosceles triangles. An isosceles triangle is composed of 2 right angles where the apothem serve as the long leg.
8 feet is divided by 2 for the short leg of 1 right triangle.
Area of right triangle = ab/2 = (9.66ft * 4ft)/2 = 38.64ft² / 2 = 19.32 ft²
There are a total of 16 right triangles in a regular octagon. (8 x 2 = 16)
Area of a regular octagon = 19.32 ft² x 16 = 309.12 Choice D.