What is the area of the marked region?



The marked figure is a quadrilateral.
It is symmetric at the square diagonal that crosses it, so it is a kite.
The symmetry axis cuts the kite into two right triangles, see .
Their height is \(\dfrac{2-\sqrt{3}}{2}\), see .
Their base is \(\dfrac{1}{2}\), see .
So the area of the marked region is \[\dfrac{2-\sqrt{3}}{4}\,.\]