A regular octagon has sides that each measure 2 units long.
A square is inscribed inside, where the vertices of the square touch the regular octagon at midpoints, as shown.
What is the area of the shaded red portion?
Drawing a perpendicular from the top right vetex of the octagon to the side of the square forms a 45-45-90 right triangle
The hypotenuse = 1/2 the side of the octagon = 1
The legs are sqrt (2) / 2
So....1/4 of the shaded area forms a trapezoid with one base of 2 one base of 2 + 2(sqrt 2) / 2 = 2 + sqrt (2)
and a height of sqrt (2) / 2
So....the area of one of these trapezoids is (1/2) height ( sum of bases) =
(1/2) sqrt (2)/2 * ( 2 + 2 + 2 sqrt (2) ) =
(1/4) sqrt (2) * ( 4 + 2 sqrt (2) )
So....the total red area = 4 times this =
sqrt (2) ( 4 + 2sqrt (2) ) =
4sqrt (2) + 4 =
4 ( 1 + sqrt (2) ) ≈ 9.657 units^2