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A regular octagon, ABCDEFGH has an area of 1 square unit. What is the area of the rectangle ABEF ?

 

 Jun 22, 2021
 #1
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Area =  2s^2  ( 1 +  sqrt 2)               where  s is the side length

 

1  =  2s^2  ( 1 + sqrt 2)

 

s^2   =      1 /  [ 2 (1  + sqrt 2)  ] 

 

Connect  HC   and draw a perpendicular  BI from B to HC

Triangle   BIC is a 45-45-90  right triangle

IC  =  s / sqrt 2   

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

 

[ ABEF  ]  = [ HCDG  ]  =       HC *  CD   =    

 

s ( 1 + sqrt (2) )  *  s =

 

s^2  ( 1 + sqrt (2) )     =  

 

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

 

1 /  2

 

cool cool cool

 Jun 23, 2021
 #2
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[ABEF] = 2[(1/2) / tan(22.5º)] = 1/2 [ABCDEFGH]

 Jun 23, 2021

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