Notes
five semi-circles around a rectangle solution

Five Semi-Circles Around a Rectangle

Solution by Pythagoras' Theorem and Area of a Circle

In the above diagram, point $E$ is the midpoint of $A B$ and so is the centre of the largest semi-circle. Point $F$ is the midpoint of $C D$, and point $G$ is the centre of the smallest semi-circle.

Let $a$ be the radius of the largest semi-circle, $b$ of the smallest, and $c$ of the three semi-circles with area $36$. Then $\frac{1}{2} \pi c^2 = 36$.

As the lengths of $A B$ and $C D$ are the same, $2 a = 2 b + 2 c$. The length of $F D$ is half this, so is $b + c$, and then the length of $F G$ is $c$. The length of $C B$ is $2 c$, and of $G E$ is $a + b$. Applying Pythagoras' theorem to triangle $E F G$ shows that:

Rearranging the identity $2a = 2 b + 2 c$ yields $c = a - b$ and so $c^2 = (a - b)^2$. So combining this with the above gives:

The total area of the purple semi-circles is therefore: