Notes
two quarter circles and a semi-circle solution

Solution to the Two Quarter Circles and a Semi-Circle Puzzle

Solution by Circle Area

With the points labelled as above, let $a$ be the length of $A B$, $b$ the length of $B C$ and $c$ half the distance of $D E$. Then $a$, $b$, and $c$ are the radii of the three circles. The diagram gives $a + b = 12$, and $a + 2 c = b$.

The shaded area is given by:

From $a + 2 c = b$, $4 c^2 = (b - a)^2 = b^2 - 2 a b + a^2$. So:

So the shaded area is given by: $\frac{1}{8} \pi 12^2 = 18 \pi$.

Solution by Invariance Principle

There are two special cases,in the first the two quarter circles are the same size and the cut out semi-circle has no area. In the second, the smaller quarter circle has no size.

The area here is $\frac{1}{2} \pi 6^2 = 18\pi$.

The area here is $\frac{1}{4}\pi 12^2 - \frac{1}{2}\pi 6^2 = 18\pi$.