10.115. On the diameter $2 R$ of a semicircle, a regular triangle is constructed, the side of which is equal to the diameter. The triangle is located on the same side of the diameter as the semicircle. Calculate the area of that part of the triangle which lies outside the circle.
Solution.

$\triangle A O D, \triangle D O E, \triangle O E C, \triangle D B E$ are equilateral triangles with side $R$ (Fig. 10.108). The desired area $S=S_{\triangle D B E}-S_{\text {segm }} . S_{\text {segm }}=\frac{\pi}{6} R^{2}-\frac{\sqrt{3}}{4} R^{2} ;$ $S_{\triangle D B E}=\frac{\sqrt{3}}{4} R^{2}, S=\frac{\sqrt{3}}{4} R^{2}-\frac{\pi}{6} R^{2}+\frac{\sqrt{3}}{4} R^{2}=R^{2}\left(\frac{3 \sqrt{3}-\pi}{6}\right)$.

Answer: $\quad R^{2}\left(\frac{3 \sqrt{3}-\pi}{6}\right)$