If the area of the regular hexagon is 90, what is the area of the purple region?
The area of a regular hexagon A is related to its length L by
\(A = \dfrac{3\sqrt 3}2 L^2\)
Consider the top white isosceles triangle.
The base is \(2L\sin 60^\circ = \sqrt 3 L\). This is also the length of the rectangle.
The width of the rectangle is \(L\), so the area of the rectangle is \(\sqrt 3 L^2 = \dfrac23 A\).
The area of rectangle is two thirds of the area of the hexagon, which is 60.