10.338. A curvilinear triangle is formed by three equal mutually tangent arcs of circles of radius $R$. Find the area of this triangle.
Solution.

The desired area is the difference between the area of the equilateral $\triangle ABC$ with side $2R$ and the sum of the areas of the sectors bounded by the given arcs (Fig. 10.123): $S=S_{\triangle ABC}-3 S_{\text{sect}}=\frac{4 R^{2} \sqrt{3}}{4}-3 \cdot \frac{\pi R^{2}}{6}=\frac{R^{2}(2 \sqrt{3}-\pi)}{2}$.

Answer: $\frac{R^{2}(2 \sqrt{3}-\pi)}{2}$.