What is the area of the marked region?



The marked area is one quarter of the complement of the circle with radius \(1\) of one region with area \(1-\dfrac{\sqrt{3}-1}{2}\), see and four regions with area \(\dfrac{2\sqrt{3}-3}{8}\), see .
Hence its area is \[\dfrac{1}{4}(\pi - 1+\dfrac{\sqrt{3}-1}{2}-4\cdot\dfrac{2\sqrt{3}-3}{8} = \dfrac{\pi}{4} - \dfrac{\sqrt{3}}{8})\,. \]