What is the area of the marked region?



The marked region is the union of two circular sectors with angle \(\dfrac{2\pi}{3}\). We are going to use the inclusion-exclusion formula.
The area of the two circular sectors is \(\dfrac{2\pi}{3}\) and the intersection of the two circular sectors is a rhombus of area \(\dfrac{\sqrt{3}}{2}\),see
Hence the area of the marked region is \[\dfrac{4\pi}{3} - \dfrac{\sqrt{3}}{2}\,.\]