What is the area of the marked region?



Consider the regions in the following picture.

The triangle is equilateral, see .
Its side length is \(1\), so its area is \(\dfrac{\sqrt{3}}{4}\).

The two circular sectors have radius \(1\).
Since the angles of the equilateral triangle are \(60^\circ\), the central angle of each circular sector is \(30^\circ\).
Thus, each circular sector is one twelfth of a circle and has area \(\dfrac{\pi}{12}\).

The area of the marked region is the area of the square subtracting the area of the triangle and the area of the two circular sectors, so it is \[1-\dfrac{\sqrt{3}}{4}-\dfrac{\pi}{6}\,.\]