What is the area of the marked region?



The marked figure is a quadrilateral.
It has a symmetry axis in common with the square, so it is a kite.
The symmetry axis cuts the kite into two right triangles, see .

One of their legs has length \(\dfrac{2\sqrt{3}-3}{6}\), see .
The other leg has length \(\dfrac{2-\sqrt{3}}{2}\), see .
The area of the marked region is then \[\dfrac{7\sqrt{3}-12}{12}\,.\]