Agardener must replace the shingles on the roof of a gazebo. the roof is a regular octagon. each side is 8 feet long, and the apothem is 9.66 feet. find the area of the roof that must be shingled. a.39 ft. / b.78 ft. / c.233 ft. / d.309 ft

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.

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