What is the area of the marked region?



The marked area is the intersection of two circular sectors.
The area of the union of those two sectors is two times the area of equilateral triangles with side length \(1\) , see , so \(\dfrac{\sqrt{3}}{2}\) and the area of one such circular sector is \(\dfrac{\pi}{6}\) because the circular sector is one sixth of a circle with radius \(1\).
Hence the area of the marked region is given by \[2\cdot \dfrac{\pi}{6} - \dfrac{\sqrt{3}}{2} = \dfrac{\pi}{3}- \dfrac{\sqrt{3}}{2}\,.\]