Theorem 1. The volume of a spherical cap is $\pi h^2\!\left(r\!-\!\frac{h}{3}\right)$ , when $h$ is its height and $r$ is the radius of the sphere.

Proof. The sphere may be formed by letting the circle $(x\!-\!r)^2\!+\!y^2 = r^2$ , i.e. $y = (\pm)\sqrt{rx\!-\!x^2}$ , rotate about the $x$ -axis. Let the spherical cap be the portion cut from the sphere on the left of the plane at $x = h$ perpendicular to the $x$ -axis.

Theorem 2. The volume of a spherical sector is $\frac{2}{3}\pi{r}^2h$ , where $h$ is the height of the spherical cap of the spherical sector and $r$ is the radius of the sphere.

Proof. The volume $V$ of the spherical sector equals to the sum or difference of the spherical cap and the circular cone depending on whether $h < r$ or $h > r$ . If the radius of the base circle of the cone is $\varrho$ , then