4.50. Calculate the solid angle of a cone with an angle of $2 \alpha$ at the vertex.
4.50. Let $O$ be the vertex of the cone, $OH$ its height. Construct a sphere of radius 1 with center $O$ and consider its section by a plane passing through the line $OH$. Let $A$ and $B$ be points on the cone lying on the sphere; $M$ be the point of intersection of the ray $OH$ with the sphere (Fig. 40). Then $HM = OM - OH = 1 - \cos \alpha$. The solid angle of the cone is equal to the area of the spherical segment cut off by the base of the cone. According to problem 4.24, this area is $2 \pi R h = 2 \pi (1 - \cos \alpha)$.