10,11

The angle in the development of the lateral surface of a cone is $120^{\circ}$. Find the angle at the vertex of the axial section of the cone.
Let $\alpha$ be the angle in the axial section of the cone, $l$ be the slant height of the cone, and $r$ be the radius of the base of the cone. The development of the lateral surface of the cone is a sector of a circle with radius $l$ (Fig.1). According to the problem, the angle between the radii of this sector is $120^{\circ}$, so the length of the arc of the sector is one-third of the circumference of a circle with radius $l$, i.e., $\frac{2 \pi l}{3}$. On the other hand, the length of the arc of the sector is equal to the circumference of the base of the cone, i.e., $2 \pi r$ (Fig.2). From the equation $\frac{2 \pi l}{3}=2 \pi r$, we find that

$$
\sin \frac{\alpha}{2}=\frac{r}{l}=\frac{1}{3}
$$

## Answer

$2 \arcsin \frac{1}{3}$.