3.43. Find the angle at the vertex of the axial section of a cone if the central angle in the development of its lateral surface is $\alpha$ radians.
3.43. Let $\triangle A B C$ be the axial section of a cone (Fig. 3.45). Then the lateral surface of the cone unfolds into a sector of a circle, with the radius $R$ equal to the slant height $B C$ of the cone; the length of the arc of the sector ($l$) is equal to the circumference of the base of the cone,

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-526.jpg?height=521&width=389&top_left_y=109&top_left_x=763)

Fig. 3.44 i.e., $l=2 \pi \cdot O C$. Using the formula $l=r \alpha$, where $r$ is the radius of the sector and $\alpha$ is the radian measure of the central angle subtending the arc of the sector. We have $2 \pi \cdot O C=B A \cdot \alpha$, hence $\frac{O C}{B A}=\frac{\alpha}{2 \pi}$. Since $\frac{O C}{B A}=\sin \angle O B C$, then $\angle O B C=\arcsin \frac{\alpha}{2 \pi}$, i.e., $\angle A B C=2 \arcsin \frac{\alpha}{2 \pi}$.