Notes
circle in a hexagon in a semi-circle solution

Solution to the Circle in a Hexagon in a Semi-Circle Puzzle

Solution by Lengths in a Regular Hexagon and Pythagoras' Theorem

In the above diagram, the point $O$ is the centre of the circle and the point $A$ is the midpoint of the top edge of the hexagon. As that top edge is a chord of the circle, its perpendicular bisector passes through the centre so $O$ is also the midpoint of the bottom edge.

Let $r$ be the radius of the shaded circle and $R$ the radius of the semi-circle, so $O A$ has length $2 r$ and $O B$ has length $R$. By considering the lengths in a regular hexagon, the length of $O A$ is $\sqrt{3}$ times the side length of the hexagon, so $A B$ has length $\frac{r}{\sqrt{3}}$. Applying Pythagoras' theorem to triangle $O A B$ shows that:

Since the area of the circle is $\pi r^2$ and of the semi-circle is $\frac{1}{2} \pi R^2$, the fraction of the semi-circle that is shaded is $\frac{6}{13}$.