2.19. The lateral surface of a cone is unfolded on a plane into a sector with a central angle of $120^{\circ}$ and an area of $S$. Find the volume of the cone.
2.19. Let $r$ be the radius of the base of the cone, and $l$ be its slant height. Then the area of the lateral surface $S=\frac{1}{2} l^{2} \cdot \frac{2 \pi}{3}=\frac{\pi l^{2}}{3}$, 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 can be found using the formula $V=\frac{1}{3} \pi r^{2} h$, where $h=\sqrt{l^{2}-r^{2}}=\sqrt{\frac{3 S}{\pi}-\frac{S}{3 \pi}}=2 \sqrt{\frac{2 S}{3 \pi}}$. Finally, we get

$$
V=\frac{1}{3} \cdot \frac{\pi S}{3 \pi} \cdot 2 \sqrt{\frac{2 S}{3 \pi}}=\frac{2 S \sqrt{6 \pi S}}{27 \pi}
$$

Answer: $\frac{2 S \sqrt{6 \pi S}}{27 \pi}$.

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-465.jpg?height=392&width=369&top_left_y=1244&top_left_x=46)

Fig. 2.18