# Nested Circles and a Triangle +-- {.image} [[NestedCirclesandaTriangle.png:pic]] > What fraction is shaded? The triangle is equilateral. =-- ## Solution by [[Lengths in an Equilateral Triangle]] +-- {.image} [[NestedCirclesandaTriangleLabelled.png:pic]] =-- 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: $$ \pi (2 r)^2 - \pi \left(\frac{3}{2} r\right)^2 + \pi r^2 = \frac{11}{4} \pi r^2 $$ The area of the outer circle is $4 \pi r^2$ so the fraction that is shaded is $\frac{11}{16}$.