What fraction of the regular octagon is shaded blue?
Call the side of the octagon, S
Area of octagon = 2(1 + sqrt (2) ) S^2
Blue area = 2 (1/2) (S/sqrt2)^2 + (S/sqrt (2))*S =
S^2 / 2 + S^2 / sqrt (2) =
( 1/2 + 1/sqrt (2)) S^2 =
(1/2) ( 1 + sqrt (2) ) S^2
Blue Area / Total Area = (1/2) / (2) = 1/4
