My semicircle

Let the center of the circle be $O$ . Then $\overline {OE}$ is perpendicular to $\overline {AB}$ . Area of $\triangle {AOC}$ is $\dfrac{1}{2}\times 2\sin (\dfrac{π}{3})\cos (\dfrac{π}{3})=\dfrac{√3}{4}$ . Area of $\triangle {OEA}$ is $\dfrac{1}{2}\tan (\dfrac{π}{3})=\dfrac{1}{2√3}$ . Therefore, area of $\triangle {OEC}$ is $\dfrac{√3}{4}-\dfrac{1}{2√3}=\dfrac{√3}{12}$ . Let $\overline {OE}$ extended meets the semicircle at $F$ . Then area of the sector $OCF$ is $\dfrac{1}{2}\times (\dfrac{π}{2}-\dfrac{π}{3})=\dfrac{π}{12}$ and the area of the region $ECF$ is $\dfrac{π-√3}{12}$ , and hence the area of the region $DEC$ (the green region) is $\dfrac{π-√3}{6}$ . Therefore $n=\boxed 3$ .