# Spherical sector

A spherical sector (blue)
A spherical sector

In geometry, a spherical sector is a portion of a sphere defined by a conical boundary with apex at the center of the sphere. It can be described as the union of a spherical cap and the cone formed by the center of the sphere and the base of the cap.

If the radius of the sphere is denoted by r and the height of the cap by h, the volume of the spherical sector is

${\displaystyle V={\frac {2\pi r^{2}h}{3}}\,.}$

This may also be written as:

${\displaystyle V={\frac {2\pi r^{3}}{3}}(1-\cos \varphi )\,,}$

where φ is half the cone angle, i.e., the angle between the rim of the cap and the direction to the middle of the cap as seen from the sphere center.

The surface area of the spherical sector -excluding the cone surface- is

${\displaystyle A=2\pi rh\,.}$

## Derivation

${\displaystyle V=\int _{0}^{2\pi }\!\int _{0}^{\varphi }\!\int _{0}^{r}\!\rho ^{2}\sin \phi \,d\rho \,d\phi \,d\theta ={\frac {2\pi r^{3}}{3}}\int _{0}^{\varphi }\!\sin \phi \,d\phi ={\frac {2\pi r^{3}}{3}}(1-\cos \varphi )\,,}$

and

${\displaystyle A=\int _{0}^{2\pi }\!\int _{0}^{\varphi }\!r\sin \phi \,d\phi \,d\theta =2\pi r\int _{0}^{\varphi }\!\sin \phi \,d\phi =2\pi r(1-\cos \varphi )\,,}$

where φ is inclination (or elevation) and θ is azimuth (right).