n regular octagon ABCDEFGH, M and N are midpoints of line BC and line FG respectively. Compute [ABMO]/[ABCDE]. ([ABCD] denotes the area of polygon ABCD.)
We can let the area of the entire octagon be 1. Then..
[ABCDE] = 1/2
area of △AOB = 1/8
area of △BOM = 1/2 of 1/8 = 1/16
[ABMO] = area of △AOB + area of △BOM = 1/8 + 1/16 = 3/16
[ABMO] / [ABCDE] = ( 3/16 )/( 1/2 ) = 3/8