Notes
a semi-circle and a quarter circle in a rectangle solution

Solution to the A Semi-Circle and a Quarter Circle in a Rectangle Puzzle

Solution by Pythagoras' Theorem and the Area of a Circle

Label the points as above, where $B$ is the centre of the semi-circle and $F$ is the point where the circles meet. Then $D F B$ is a straight line.

Let $r$ be the radius of the semi-circle, so the vertical side of the rectangle has length $2r$, and this is then the radius of the quarter circle. In terms of $r$, the shaded area is:

Triangle $B C D$ is right-angled, so it satisfies Pythagoras' theorem. Its side lengths are $4$, $r$, and $3r$, so:

so $8 r^2 = 16$ and hence $r^2 = 2$.

The shaded area is therefore $3 \pi$.