What is the area of the marked region?



The marked region can be obtained from the square by subtracting the following regions:
four regions whose area is \(1-\dfrac{\sqrt{3}}{4}-\dfrac{\pi}{6}\), see
and four regions whose area is \(\dfrac{\sqrt{3}}{2}+\dfrac{\pi}{12}-1\), see

We deduce that the area of the marked region is \[1- 4\cdot (1-\dfrac{\sqrt{3}}{4}-\dfrac{\pi}{6}) -4\cdot (\dfrac{\sqrt{3}}{2}+\dfrac{\pi}{12}-1) = 1-\sqrt{3}+\dfrac{\pi}{3}\,.\]