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how. 

 Jun 28, 2018
 #1
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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

 

 

cool cool cool

 Jun 28, 2018
edited by CPhill  Jun 28, 2018

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