What is the area of the marked region?



The circle has area \(\pi\).
The marked region is the complement in the circle of a region with area \(\dfrac{2\pi}{3} - \dfrac{\sqrt{3}}{2}\),see .
Hence the area of the marked region is \[\pi - \dfrac{2\pi}{3} + \dfrac{\sqrt{3}}{2} = \dfrac{\pi}{3} + \dfrac{\sqrt{3}}{2}\,.\]