# Solution to the Square Overlapping a Circle Puzzle +-- {.image} [[SquareOverlappingaCircle.png:pic]] > A square, and a circle of radius $2$. What's the total shaded area? =-- ## Solution by [[Intersecting Chords Theorem]] +-- {.image} [[SquareOverlappingaCircleLabelled.png:pic]] =-- 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 triangle|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 $2$.