What is the area of the marked region?



The circle has radius \(1\) hence area \(\pi\).
The square and the four triangles form a region with area \[4-2\sqrt{3}+4\cdot(\sqrt{3} -\frac{3}{2})=2\sqrt{3}-2\] see and .
The complement of this region in the circle consists in four copies of the marked region.
So the area of the marked region is \[\dfrac{1}{4}\left(\pi - (2\sqrt{3}-2) )\right) = \dfrac{\pi}{4} + \dfrac{1}{2} - \dfrac{\sqrt{3}}{2}\,. \]