A regular octagon, ABCDEFGH has an area of 1 square unit. What is the area of the rectangle ABEF ?
+129845 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
