Notes
two semi-circles and a rectangle solution

Solution to the Two Semi-Circles and a Rectangle Puzzle

Solution by the Intersecting Chords Theorem

With the points labelled as above, let $a$ be the length of $A P$ and $b$ the radius of the smaller semi-circle, so the length of $B P$. The area of the rectangle is then $a b$. Applying the intersecting chords theorem to $A P$ and $C P$ shows that $a b = 8^2$. So the area of the rectangle is $64$.