Notes
three squares overlapping a semi-circle solution

Solution to the Three Squares Overlapping a Semi-Circle Puzzle

Solution by Intersecting Chords Theorem

As $A P$ is at right-angles to the diameter of the semi-circle, it is a tangent to it at $A$. The intersecting chords theorem then says that:

Each small square has area $4$ so has side length $2$. Therefore $P C$ has length $2$ and $P D$ length $4$. Hence the area of the red square is $2 \times 4 = 8$.