What is the area of the marked region?



The marked region is an octagon.
All sides have length \(\dfrac{\sqrt{3}-1}{2}\),see .
The angles in the octagon alternate between \(120^\circ\) and \(150^\circ\).


The octagon is a square without four congruent triangles.
These triangles have area \(\dfrac{7\sqrt{3}-12}{24}\) , see .
The square has a side length \[\dfrac{1}{2} - \dfrac{2-\sqrt{3}}{2} = \dfrac{\sqrt{3}-1}{2}\] see ,and .
So the area of the octagon is \[ \left(\dfrac{\sqrt{3}-1}{2}\right)^2 - 4\cdot \dfrac{7\sqrt{3}-12}{24} = 3-\dfrac{5\sqrt{3}}{3}\,.\]