Notes
circle and two squares solution

Solution to the Circle and Two Squares Puzzle

Solution by Congruency

In the above diagram, the point labelled $O$ is the centre of the circle. Since the angle in a semi-circle is a right-angle, $A E$ is a diameter of the circle, and so passes through $O$ as shown.

Triangle $A D C$ is obtained by rotating triangle $A F E$ by $90^\circ$ clockwise, so $A C$ is the continuation of $A B$. Since the length of $A B$ is the same as a radius of the circle, the rotation takes $O$ to $B$. Therefore, the line segment $O F$ is taken to $B D$ and so they have the same length. The circle therefore has radius $2$ and area $4 \pi$.