- Source: 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
=
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 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
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 sector (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