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A regular octagon has sides that each measure 2 units long.

A square is inscribed inside, where the vertices of the square touch the regular octagon at midpoints, as shown.

What is the area of the shaded red portion?

 

 Jan 11, 2021
 #1
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Drawing a perpendicular  from the  top right  vetex of the  octagon  to the side of the  square  forms a 45-45-90  right triangle

The hypotenuse  =   1/2  the side of the octagon  = 1

The  legs  are   sqrt (2)   / 2

 

So....1/4 of the shaded area  forms a trapezoid with   one  base of 2    one base of 2 + 2(sqrt 2) / 2  = 2 + sqrt (2)

and a height of  sqrt (2) / 2

 

So....the area of one of these trapezoids  is    (1/2)  height ( sum of bases)  =

 

(1/2)  sqrt (2)/2   *  ( 2 + 2 + 2 sqrt (2) )   =

 

(1/4) sqrt (2)  * ( 4 + 2 sqrt (2) )

 

So....the total red area  =  4 times this  = 

 

sqrt (2)  ( 4 + 2sqrt (2) )  =

 

4sqrt (2)  + 4  =

 

4 ( 1 + sqrt (2) )  ≈   9.657  units^2

 

 

cool cool cool

 Jan 11, 2021

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