What is the area of the marked region?



The regular hexagon consists of six equilateral triangles with side length \(1\), each with area \(\dfrac{\sqrt{3}}{4}\).
The marked region is repeated six times in the hexagon.
The six regions together are the complement in the hexagon of six regions with area \(\dfrac{\pi}{3}-\dfrac{\sqrt{3}}{2}\), see .
So the area of the marked region is \[\frac{1}{6}\Big(6\dfrac{\sqrt{3}}{4}-6\Big(\dfrac{\pi}{3}-\dfrac{\sqrt{3}}{2}\Big)\Big)= \dfrac{3\sqrt{3}}{4}-\dfrac{\pi}{3}\,.\]