Spherical sector

A spherical sector (blue)

A 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.

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

This may also be written as $${\displaystyle V={\frac {2\pi r^{3}}{3}}(1-\cos \varphi )\,,}$$ 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 limiting case is for φ approaching 180 degrees, which then describes a complete sphere.

The height, h is given by $${\displaystyle h=r(1-\cos \varphi )\,.}$$

The volume V of the sector is related to the area A of the cap by: $${\displaystyle V={\frac {rA}{3}}\,.}$$

Area

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

It is also $${\displaystyle A=\Omega r^{2}}$$ 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 A = r².

Derivation

Further information: double integral and triple integral

The volume can be calculated by integrating the differential volume element $${\displaystyle dV=\rho ^{2}\sin \phi \,d\rho \,d\phi \,d\theta }$$ over the volume of the spherical sector, $${\displaystyle V=\int _{0}^{2\pi }\int _{0}^{\varphi }\int _{0}^{r}\rho ^{2}\sin \phi \,d\rho \,d\phi \,d\theta =\int _{0}^{2\pi }d\theta \int _{0}^{\varphi }\sin \phi \,d\phi \int _{0}^{r}\rho ^{2}d\rho ={\frac {2\pi r^{3}}{3}}(1-\cos \varphi )\,,}$$ 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 $${\displaystyle dA=r^{2}\sin \phi \,d\phi \,d\theta }$$ over the spherical sector, giving $${\displaystyle A=\int _{0}^{2\pi }\int _{0}^{\varphi }r^{2}\sin \phi \,d\phi \,d\theta =r^{2}\int _{0}^{2\pi }d\theta \int _{0}^{\varphi }\sin \phi \,d\phi =2\pi r^{2}(1-\cos \varphi )\,,}$$ where φ is inclination (or elevation) and θ is azimuth (right). Notice r is a constant. Again, the integrals can be separated.

See also

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

References

1. Weisstein, Eric W. "Spherical sector". MathWorld .
2. Weisstein, Eric W. "Spherical cone". MathWorld .