Notes
semi-circle on a hexagon solution

Solution to the Semi-Circle on a Hexagon Puzzle

Solution by Area of a Trapezium and Semi-circle, Similarity, and Properties of a Regular Hexagon

In the above diagram, the point labelled $I$ lies on $F C$ below $E$ so that angle $F \hat{I} E$ is a right-angle. Since $I$ is the midpoint of $F O$, which has length $4$, $F I$ has length $2$.

Let $x$ be the length of $G J$, so $G H$ has length $2 x + 4$, and let $h$ be the length of $E I$. Then as triangles $F I E$ and $G J E$ are similar, $E J$ has length $\frac{x h}{2}$. From the formula for the area of a trapezium, the shaded region has area

Since $h$ is the height of one of the six equilateral triangles making up the hexagon, the area of the is $6 \times \frac{1}{2} \times 4 h = 12 h$. So since the shaded area is one third of the hexagon:

which simplifies to $x(4 + x) = 8$.

The radius of the semi-circle is $2 + x$ so its area is

From above, $4 x + x^2 = x(4 + x) = 8$, so the area simplifies to: