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.
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
This may also be written as
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:
The curved surface area of the spherical sector (on the surface of the sphere, excluding the cone surface) is
It is also
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 A = r2.