Notes
semi-circle splitting a triangle solution

Semi-Circle Splitting a Triangle

Solution by Area Scale Factor and Lengths in Equilateral Triangles

With the points labelled as above, the orange triangle has half the area of the full triangle, so the length of $E F$ is $\frac{1}{\sqrt{2}}$ of the length of $A B$, so is $4\sqrt{2}$. The height $O C$ is then $\frac{\sqrt{3}}{2}$ times the length of $E F$, so is $\frac{\sqrt{3}}{2} \times 4 \sqrt{2} = 2\sqrt{6}$. The area of the semi-circle is therefore $\frac{1}{2} \pi (2 \sqrt{6})^2 = 12 \pi$.