Spherical sector explained

In geometry, a spherical sector, also known as a spherical cone, 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

If the radius of the sphere is denoted by and the height of the cap by, the volume of the spherical sector is$$V\; =\; \backslash frac\backslash ,.$$

This may also be written as$$V\; =\; \backslash frac\; (1-\backslash cos\backslash varphi)\backslash ,,$$where is half the cone angle, i.e., 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 limiting case is for approaching 180 degrees, which then describes a complete sphere.

The height, is given by$$h\; =\; r\; (1-\backslash cos\backslash varphi)\backslash ,.$$

The volume of the sector is related to the area of the cap by:$$V\; =\; \backslash frac\backslash ,.$$

Area

The curved surface area of the spherical sector (on the surface of the sphere, excluding the cone surface) is$$A\; =\; 2\backslash pi\; rh\backslash ,.$$

It is also $$A\; =\; \backslash Omega\; r^2$$where 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 .

Derivation

The volume can be calculated by integrating the differential volume element$$dV\; =\; \backslash rho^2\; \backslash sin\; \backslash phi\; \backslash ,\; d\backslash rho\; \backslash ,\; d\backslash phi\; \backslash ,\; d\backslash theta$$over the volume of the spherical sector,$$V\; =\; \backslash int\_0^\; \backslash int\_0^\backslash varphi\backslash int\_0^r\backslash rho^2\backslash sin\backslash phi\; \backslash ,\; d\backslash rho\; \backslash ,\; d\backslash phi\; \backslash ,\; d\backslash theta\; =\; \backslash int\_0^\; d\backslash theta\; \backslash int\_0^\backslash varphi\; \backslash sin\backslash phi\; \backslash ,\; d\backslash phi\; \backslash int\_0^r\; \backslash rho^2\; d\backslash rho\; =\; \backslash frac\; (1-\backslash cos\backslash varphi)\; \backslash ,,$$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 $$dA\; =\; r^2\; \backslash sin\backslash phi\; \backslash ,\; d\backslash phi\; \backslash ,\; d\backslash theta$$over the spherical sector, giving$$A\; =\; \backslash int\_0^\; \backslash int\_0^\backslash varphi\; r^2\; \backslash sin\backslash phi\; \backslash ,\; d\backslash phi\; \backslash ,\; d\backslash theta\; =\; r^2\; \backslash int\_0^\; d\backslash theta\; \backslash int\_0^\backslash varphi\; \backslash sin\backslash phi\; \backslash ,\; d\backslash phi\; =\; 2\backslash pi\; r^2(1-\backslash cos\backslash varphi)\; \backslash ,,$$where is inclination (or elevation) and is azimuth (right). Notice is a constant. Again, the integrals can be separated.

See also

• Circular sector — the analogous 2D figure.
• Spherical cap
• Spherical segment
• Spherical wedge