In geometry, a spherical sector, also known as a spherical cone, is a portion of a ball that is bounded by a spherical cap and the cone that connects the centre of the sphere to the boundary 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 = 2 π r 2 h 3 . {\displaystyle V={\frac {2\pi r^{2}h}{3}}\,.}
This may also be written as
V = 2 π r 3 3 ( 1 − cos φ ) , {\displaystyle V={\frac {2\pi r^{3}}{3}}(1-\cos \varphi )\,,}
where φ is half the cone aperture angle, i.e., φ is the angle between the rim of the cap and the axis 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
h = r ( 1 − cos φ ) . {\displaystyle h=r(1-\cos \varphi )\,.}
The volume V of the sector is related to the area A of the cap by:
V = r A 3 . {\displaystyle V={\frac {rA}{3}}\,.}
Area The curved surface area of the spherical cap (on the surface of the sphere, excluding the cone surface) is
A = 2 π r h . {\displaystyle A=2\pi rh\,.}
It is also
A = Ω r 2 {\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
The volume can be calculated by integrating the differential volume element
d V = ρ 2 sin ϕ d ρ d ϕ d θ {\displaystyle dV=\rho ^{2}\sin \phi \,d\rho \,d\phi \,d\theta }
over the volume of the spherical sector,
V = ∫ 0 2 π ∫ 0 φ ∫ 0 r ρ 2 sin ϕ d ρ d ϕ d θ = ∫ 0 2 π d θ ∫ 0 φ sin ϕ d ϕ ∫ 0 r ρ 2 d ρ = 2 π r 3 3 ( 1 − cos φ ) , {\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
d A = r 2 sin ϕ d ϕ d θ {\displaystyle dA=r^{2}\sin \phi \,d\phi \,d\theta }
over the spherical sector, giving
A = ∫ 0 2 π ∫ 0 φ r 2 sin ϕ d ϕ d θ = r 2 ∫ 0 2 π d θ ∫ 0 φ sin ϕ d ϕ = 2 π r 2 ( 1 − cos φ ) , {\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