#1**+3 **

We have an octagon with an apothem of 8

We have a right triangle formed

The central angle formed by the triangle is 22.5°

So...the base of this triangle (1/2 side length of the octagon) is given by 8 tan (22.5°)

The apothem and the base therefore form two legs of the right triangle

And the area of a right triangle = (1/2) product of the leg lengths

So....the area of the right triangle is (1/2) (8) (8 tan 22.5°)

But we have 16 of these congruent right triangles that comprise the area of the octagon

So...the total area is (16) (1/2) (8) * 8 tan (22.5°) =

8^3 * tan (22.5°) ≈ 212.1 units^2

CPhill
Jun 28, 2018