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.

Volume

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

Area

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 by a spherical sector with A = r².

Derivation

The volume can be calculated by integrating the differential volume element

dV=\rho ^{2}\sin \phi d\rho d\phi d\theta

over the volume of the spherical sector,

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

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}\sin \phi d\phi d\theta

over the spherical sector, giving

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

where φ is inclination (or elevation) and θ is azimuth (right). Notice r is a constant. Again, the integrals can be separated.

External links

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Spherical sector

Volume

Area

Derivation

See also

External links