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A square is inscribed in a regular dodecagon--a regular polygon with 12 sides--so that they share 4 vertices, as shown.  If the area of the dodecagon is 12, what is the area of the square?

 

 Jan 7, 2021
 #1
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If we  inscribed  the  dodecagon  in a circle  we  can  find   the radius  as

 

1 = (1/2)R^2  ( sin 360 /12)

1  =  (1/2) R^2  ( sin 30°)

1  =  (1/4) R^2

4  = R^2

2   = R  of  circle

 

Diagonal of  square =  4

 

Side of square =   4/sqrt (2)

 

Area of square =   (4/sqrt (2))^2   =  16/2   =   8

 

cool cool cool

 Jan 7, 2021

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