What is the area of the marked region?



The big circle has area \(\pi\).
The marked region is replicated four times in the big circle.
These four regions together are the complement in the circle of:
four regions with area \(\dfrac{\pi}{8} - \dfrac{1}{4}\), see , and four regions with area \(\dfrac{1}{2}\), see .
Hence the area of the marked region is \[\dfrac{1}{4}\left(\pi- 4\cdot \left(\dfrac{\pi}{8} - \dfrac{1}{4}\right) - 4\cdot \dfrac{1}{2}\right) =\dfrac{\pi}{8} - \dfrac{1}{4}\,. \]