# Solution to the Circle In Pentagon in Semi-Circle Puzzle +-- {.image} [[CircleInPentagoninSemiCircle.png:pic]] > A circle in a regular pentagon in a semicircle. What fraction of the semicircle is shaded? =-- ## Solution by [[Lengths in a Regular Pentagon]] +-- {.image} [[CircleInPentagoninSemiCircleLabelled.png:pic]] =-- The radius of the small circle is the radius of the [[incircle]] of the [[regular pentagon]]. The radius of the semi-circle is the radius of the incircle added to the radius of the [[circumcircle]] of the regular pentagon. From [[lengths in a regular pentagon]], these radius of the circumcircle, $R$, and incircle, $r$, are related by $2 r = \phi R$ where $\phi$ is the [[golden ratio]]. With this notation, the radius of the semi-circle is $r + R = (1 + \frac{2}{\phi}) r$. The area of the incircle is $\pi r^2$ and the area of the semi-circle is: $$ \begin{aligned} \frac{1}{2} \pi (r + R)^2 &= \frac{1}{2} \pi (1 + \frac{2}{\phi})^2 r^2 \\ &= \frac{1}{2 \phi^2} \pi (\phi + 2)^2 r^2 \\ &=\frac{1}{2 \phi^2} \pi r^2 (\phi^2 + 4 \phi + 4) \\ &=\frac{1}{2 \phi^2} \pi r^2 5 \phi^2 \\ &= \frac{5}{2} \pi r^2 \end{aligned} $$ Therefore the fraction of the semi-circle that is shaded is $\frac{2}{5}$.