64. Find the value of the solid angle of a circular cone with an angle a at the vertex.
64. Consider a circular cone with vertex $S$, axis $S O$ and angle $z$ at the vertex. Take any point $O$ on the axis $S O$ inside the cone and drop a perpendicular from $O$ to any of the generators (Fig. 125). All other perpendiculars to the generators can be obtained by rotating the constructed perpendicular around the axis $S O$. Therefore, all perpendiculars to the generators lie on a cone with vertex $O$, axis $O S$ and angle at the vertex $\pi-\%$. Inscribed in the cone $S$ is a regular polyhedral angle $S_{n}$. Let its faces intersect the cone $S$ along the generators $S A_{1}, S A_{3}, \ldots, S A_{n}$.

Then construct a regular polyhedral angle $O_{n}$, the faces of which will touch the cone $O$ along the generators $O A_{1}$, $O A_{2}, \ldots, O A_{n}$.

It is easy to see that the polyhedral angle $O A_{1} A_{2} \ldots A_{n}$ will be symmetric to the supplementary angle of the polyhedral angle $S_{n}$, since its edges are perpendicular to the faces of the angle $S_{n}$, but directed in the opposite direction.

By the previous problem, $\mu S_{n}=\operatorname{def} O_{n}$, where by $O_{n}$ we denote the polyhedral angle $O A_{1} A_{2} \ldots A_{n}$. If we increase $n$ indefinitely, then $S_{n}$ will get closer and closer
to the cone $S$, and $O_{n}$ to the cone $O$. Therefore, passing to the limit, we will have $\mu S=\operatorname{def} O$, meaning by $\operatorname{def} O$ the deficiency of the angle of the cone's development to $2 \pi$.

If the angle at the vertex of the cone $S$ is $\alpha$, then the angle at the vertex of the cone $O$ is $\pi-\alpha$. Let the generator of the cone $O$

![](https://cdn.mathpix.com/cropped/2024_05_21_d3653343768e91d6de9ag-154.jpg?height=564&width=574&top_left_y=418&top_left_x=315)

Fig. 125.

be $l$, then the lateral surface area of the cone $O$ is $\pi l^{2} \sin \frac{\pi-x}{2}$. On the other hand, the area of the development of the lateral surface of the cone $O$ is $\frac{\varphi}{2} l^{2}$, where $\varphi$ is the development angle in radians. Therefore, $\varphi=2 \pi \sin \frac{\pi-a}{2}$. Consequently,

$$
\mu S=\operatorname{def} O=2 \pi-\vartheta=2 \pi\left(1-\cos \frac{a}{2}\right)=4 \pi \text { siti } \frac{a}{4}
$$