What is the area of the marked region?



The region is a quarter of the complement of the circle with area \(\pi\) of four regions with area \(\dfrac{\pi}{8} - \dfrac{1}{4}\), see and and four regions with area \(\dfrac{1}{2}\),see .
Hence its area is \[\dfrac{1}{4}\left(\pi-\left(4\cdot \left(\dfrac{\pi}{8} - \dfrac{1}{4}\right) - \dfrac{4}{2}\right)\right) =\dfrac{\pi}{8} - \dfrac{1}{4}\,. \]