C2. A circle lies within a rectangle and touches three of its edges, as shown.
The area inside the circle equals the area inside the rectangle but outside the circle.

What is the ratio of the length of the rectangle to its width?
Solution
Let the radius of the circle be $r$ and the length of the rectangle be $s$.
Therefore the width of the rectangle is $2 r$.
The area inside the circle is $\pi r^{2}$.
The area inside the rectangle but outside the circle is $2 r s-\pi r^{2}$.
These quantities are equal so
$$
\pi r^{2}=2 r s-\pi r^{2} \text {. }
$$

Adding $\pi r^{2}$ to each side gives
$$
2 \pi r^{2}=2 r s .
$$

Since $r$ is not zero, division by $2 r$ is permissible and gives
$$
\pi r=s .
$$

Therefore the ratio of the length of the rectangle to the width is $\pi r: 2 r$, which simplifies to $\pi: 2$.