Spherical sector

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

( 1 − cos ⁡ φ )

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

The limiting case is for φ approaching 180 degrees, which then describes a complete sphere.

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

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.

The volume can be calculated by integrating the differential volume element

sin ⁡ ϕ

over the volume of the spherical sector,

sin ⁡ ϕ

where the integrals have been separated, because the integrand can be separated into a product of functions each with one dummy variable.

The area can be similarly calculated by integrating the differential spherical area element

sin ⁡ ϕ

over the spherical sector, giving

sin ⁡ ϕ

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

Notice r is a constant.

Again, the integrals can be separated.

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