Notes
square overlapping a circle solution

Solution to the Square Overlapping a Circle Puzzle

Solution by Intersecting Chords Theorem

In the diagram above, the point labelled $O$ is the centre of the circle. The area of the shaded region is given by multiplying the length of one of the sides of the square by the length of $C D$.

The intersecting chords theorem states that $C D \times A D = E D^2$, so the shaded region has area equal to the square of the length of $E D$. By symmetry, line segment $O A$ lies on a diagonal of the square, so triangle $O B A$ is isosceles and right-angled. Therefore the square of the length of $O B$ is half the square of the length of $O A$, which is $2$. Since $O B$ and $E D$ are congruent, the shaded region thus has area $2$.