10.165. A circle is inscribed in a regular triangle, and another circle is circumscribed around it. Find the area of the resulting ring if the side of the triangle is $a$.
Solution.

We have $S_{\Delta}=\frac{\sqrt{3}}{4} a^{2}, p=\frac{3}{2} a, r=\frac{S_{\Delta}}{p}=\frac{\sqrt{3}}{4} a^{2} \cdot \frac{2}{3 a}=\frac{\sqrt{3}}{6} a, R=\frac{a^{3}}{4 S_{\Delta}}=$ $=\frac{a^{3}}{\sqrt{3} a^{2}}=\frac{\sqrt{3}}{3} a$. Therefore, $S=\pi R^{2}-\pi r^{2}=\pi a^{2}\left(\frac{1}{3}-\frac{1}{12}\right)=\frac{\pi a^{2}}{4}$.

Answer: $\quad \frac{\pi a^{2}}{4}$.

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-633.jpg?height=498&width=324&top_left_y=93&top_left_x=96)

Fig. 10.138