Notes
overlapping rectangle and semi-circle solution

Solution to the Overlapping Rectangle and Semi-Circle Puzzle

Solution by Pythagoras' Theorem

In the above diagram, the point labelled $O$ is the centre of the semi-circle. Let $r$ be the radius of the semi-circle, let $a$ be the length of $O D$ and $b$ of $C D$. The length of $A D$ is is $r + a$ so the area of the rectangle is $r(r + a) = r^2 + a r$.

Applying Pythagoras' theorem to triangle $O D C$ gives the relationship:

Applying it to triangle $C D A$ gives:

Therefore the area of the rectangle is $18$.