11.186. The height of the cone is $h$. The lateral surface of this cone unfolds into a sector with a central angle of $120^{\circ}$. Calculate the volume of the cone.
## Solution.

Let $R$ be the radius of the base of the cone, and $l$ be its slant height. The length of the arc of the lateral surface development is $2 \pi R$ or $\frac{2 \pi l}{3}$. Therefore, $2 \pi R = \frac{2}{3} \pi l$, which gives $l = 3R$. Since $l^2 = R^2 + h^2$, we have $9R^2 = R^2 + h^2$, leading to $R^2 = \frac{h^2}{8}$. The volume of the cone $V = \frac{1}{3} \pi R^2 h = \frac{1}{3} \pi \cdot \frac{h^2}{8} \cdot h = \frac{\pi h^3}{24}$.

Answer: $\quad \frac{\pi h^3}{24}$.