11.101. The lateral surface of a cone is unfolded onto a plane as a sector with a central angle of $120^{\circ}$ and an area of $S$. Find the volume of the cone.
## Solution.

Let the radius of the base of the cone be $r$, and its slant height be $l$. Then the area of the development $S=\frac{1}{2} l^{2} \cdot \frac{2 \pi}{3}=\frac{\pi l^{2}}{3}$, from which $l=\sqrt{\frac{3 S}{\pi}}$. But $S=\pi r l=\pi r \sqrt{\frac{3 S}{\pi}}$ and, therefore, $r=\sqrt{\frac{S}{3 \pi}}$. The volume of the cone will be found using the formula $V=\frac{1}{3} \pi r^{2} h$, where $h=\sqrt{l^{2}-r^{2}}=\sqrt{\frac{3 S}{\pi}-\frac{S}{3 \pi}}=2 \sqrt{\frac{2 S}{3 \pi}}$. Thus, $V=\frac{1}{3} \cdot \frac{\pi S}{3 \pi} \cdot 2 \sqrt{\frac{2 S}{3 \pi}}=\frac{2 S \sqrt{6 \pi S}}{27 \pi}$.

Answer: $2 S \sqrt{6 \pi S} /(27 \pi)$.