The polygon is a regular octagon. Each shaded triangle is isosceles, has vertex angle of 360°/8 = 45°, base angles 67.5°, and base = 8.
Height of triangle = 4tan(67.5°)
area of triangle = 0.5(base)(height) = 0.5(8)(4tan(67.5°))
Shaded region = 2*0.5(8)(4tan(67.5°)) = 32tan(67.5°) ≅180.71 cm^2
