7. The area of an equilateral triangle inscribed in a circle is 81 cm $^{2}$. Find the radius of the circle.
Solution: $S_{\Delta}=81 c M^{2}, S_{\Delta}=\frac{1}{2} a \cdot h=\frac{1}{2} a \cdot a \cos 30^{\circ}=\frac{a^{2} \sqrt{3}}{4}, r=$ ?

$S_{\triangle}=3 S_{\triangle A O C}=3 \cdot \frac{1}{2} a \cdot r \cos 60^{\circ}=\frac{3}{4} a r=\frac{3}{2} r \cdot r \cdot \sin 120^{\circ}=\frac{3 r^{2} \sqrt{3}}{4}$,

$\frac{3 r^{2} \sqrt{3}}{4}=81, r^{2}=\frac{81 \cdot 4}{3 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}=36 \sqrt{3}, r=6 \sqrt[4]{3}$

![](https://cdn.mathpix.com/cropped/2024_05_06_8eeeb8d6ace591217257g-07.jpg?height=319&width=302&top_left_y=326&top_left_x=1531)

Answer: $r=6 \sqrt[4]{3}$.