Notes
nested circles and a triangle solution

Nested Circles and a Triangle

Solution by Lengths in an Equilateral Triangle

In the above diagram, $O B$ is a radius of the smallest circle, $O A$ of the largest circle, and $C B$ is a diameter of the middle circle. Since $O C$ is also a radius of the largest circle, the length of $C B$ is the sum of the lengths of $O B$ and $O A$.

From considering the lengths in an equilateral triangle, $O B$ has half the length of $O A$ and so is one third of the length of $C B$. Let $r$ be the length of $O B$, then $O A$ has length $2 r$ and the radius of the middle circle is $\frac{3}{2} r$. The shaded region then has area:

The area of the outer circle is $4 \pi r^2$ so the fraction that is shaded is $\frac{11}{16}$.