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 [math]\displaystyle{ V = \frac{2\pi r^2 h}{3}\,. }[/math]
This may also be written as [math]\displaystyle{ V = \frac{2\pi r^3}{3} (1-\cos\varphi)\,, }[/math] 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: [math]\displaystyle{ V = \frac{rA}{3}\,. }[/math]
Area
The curved surface area of the spherical sector (on the surface of the sphere, excluding the cone surface) is [math]\displaystyle{ A = 2\pi rh\,. }[/math]
It is also [math]\displaystyle{ A = \Omega r^2 }[/math] 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
The volume can be calculated by integrating the differential volume element [math]\displaystyle{ dV = \rho^2 \sin \phi \, d\rho \, d\phi \, d\theta }[/math] over the volume of the spherical sector, [math]\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) \, , }[/math] 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 [math]\displaystyle{ dA = r^2 \sin\phi \, d\phi \, d\theta }[/math] over the spherical sector, giving [math]\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) \, , }[/math] 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
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