What is the area of the marked region?



The marked region is the union of two circular sectors that overlap in a triangle.

The triangle is equilateral, see .
Its side length is \(1\) so its area is \(\dfrac{\sqrt{3}}{4}\).
The circular sectors have radius \( 1\) and central angle \(30^\circ\), so their area is \(\dfrac{\pi}{6}\).
The area of the marked region is then \[\dfrac{\pi}{6}+\dfrac{\pi}{6}-\dfrac{\sqrt{3}}{4}=\dfrac{\pi}{3}-\dfrac{\sqrt{3}}{4}\,.\]