What is the area of the marked region?



The marked region is the complement in a square of four congruent triangles .
These triangles are right triangles with area \(\dfrac{7\sqrt{3}-12}{12}\) , see .
The square has a side length \(\dfrac{1}{2} - \dfrac{2-\sqrt{3}}{2} = \dfrac{\sqrt{3}-1}{2}\),see ,and .
hence an area of \(\left(\dfrac{\sqrt{3}-1}{2}\right)^2 = \dfrac{2-\sqrt{3}}{2}\).
So the area of the octagon is \[ \dfrac{2-\sqrt{3}}{2} - 4\cdot \dfrac{7\sqrt{3}-12}{12} = \dfrac{30-17\sqrt{3}}{6} \,.\]