I saw this on the YouTube channel Math with Omani but I've not looked at the solution there. The puzzle is to find the shaded area, given on the marked length.
Scroll down to find my solution
Solution
There are probably different ways to tackle this but those right angle suggest Pythagoras is the way to go, so we draw in some more lines and create a right-angled triangle. Let the side of the square be and the radius of the circle .
We can compute as and we see the legs of triangle are and , and its hypotenuse . By Pythagoras we have
.
Substituting for gives
.
and after expanding and tidying up we get
, which factorises
and the roots are and . The first is invalid and so we conclude whence and the area is . In fact has the proportions of a 3-4-5 triangle, which is rather neat. 
