Spherical_sector Knowpia

In geometry, a spherical sector,^[1] also known as a spherical cone,^[2] is a portion of a sphere or of a ball 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. It is the three-dimensional analogue of the sector of a circle.

Volume edit

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

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

This may also be written as

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

where

φ

is half the cone angle, i.e.,

φ

is the angle between the rim of the cap and the direction to the middle of the cap as seen from the sphere center.

The volume $V$ of the sector is related to the area $A$ of the cap by:

V={\frac {rA}{3}}\,.

Area edit

The curved surface area of the spherical sector (on the surface of the sphere, excluding the cone surface) is

A=2\pi rh\,.

It is also

A=\Omega r^{2}

where

Ω

is the solid angle of the spherical sector in steradians, the SI unit of solid angle. One steradian is defined as the solid angle subtended by a cap area of