Spherical sector



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 $r$ 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., $φ$ 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: $$V = \frac{rA}{3}\,.$$

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 as the solid angle subtended by a cap area of $A = r^{2}$.

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.