Notes
square overlapping a quarter circle solution

Solution to the Square Overlapping a Quarter Circle Puzzle

Solution by Area of a Triangle and Pythagoras' Theorem

With the points labelled as above, let $x$ be the length of $O A$ so the area of the square is $x^2$. As the coloured sections have the same area, the area of triangle $O A B$ is a third of that of the square, so from the area of a triangle, $A B$ has length $\frac{2}{3} x$. Since $O B$ is a radius of the circle, it has length $13$ and so applying Pythagoras' theorem to $O A B$ gives:

This shows that the area of the square is $x^2 = 9 \times 13 = 117$.