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In geometry, a spherical sector is a portion of a sphere 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.

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., the angle between the rim of the cap and the direction to the middle of the cap as seen from the sphere center.

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. This formula can be used to define the steradian, the SI unit of solid angle. By taking a spherical sector such that A = r², there is one unit of solid angle inside it.

Derivation

The volume can be calculated by integrating the differential volume element

dV=\rho ^{2}\sin \theta 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.

External links

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@@ Line 2: / Line 2: @@
 [[File:Spherical sector.png|thumb|A spherical sector]]
 In [[geometry]], a '''spherical sector''' is a portion of a [[sphere]] 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.
+==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
@@ Line 7: / Line 9: @@
 :<math>V=\frac{2\pi r^2h}{3}\,.</math>
-This may also be written as:
+This may also be written as
 :<math>V=\frac{2\pi r^3}{3}(1-\cos\varphi)\,,</math>
@@ Line 13: / Line 15: @@
 where ''φ'' is half the [[cone (geometry)|cone]] angle, i.e., the angle between the rim of the cap and the direction to the middle of the cap as seen from the sphere center.
+==Area==
-The [[surface area]] of the spherical sector -excluding the cone surface- is
+The curved [[surface area]] of the spherical sector (on the surface of the sphere, excluding the cone surface) is
 :<math>A=2\pi rh\,.</math>
+It is also
+:<math>A=\Omega r^2  </math>
+where Ω is the [[solid angle]] of the spherical sector. This formula can be used to define the [[steradian]], the SI unit of solid angle. By taking a spherical sector such that ''A'' = ''r''<sup>2</sup>, there is one unit of solid angle inside it.
 == Derivation ==
-:<math>V=\int_0^{2\pi}\!\int_0^\varphi\!\int_0^r\!\rho^2\sin\phi\,d\rho\,d\phi\,d\theta=\frac{2\pi r^3}{3}\int_0^\varphi\!\sin\phi\,d\phi=\frac{2\pi r^3}{3}(1-\cos\varphi)\,,</math>
+{{further|double integral|triple integral}}
-and
+The volume can be calculated by integrating the [[differential volume element]]
+:<math> dV = \rho^2 \sin \theta d\rho d\phi d\theta </math>
+over the volume of the spherical sector,
+:<math> 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^2d\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>dA = r^2 \sin\phi d\phi d\theta </math>
+over the spherical sector, giving
-:<math>A=\int_0^{2\pi}\!\int_0^\varphi\! r\sin\phi\,d\phi\,d\theta=2\pi r\int_0^\varphi\!\sin\phi\,d\phi=2\pi r(1-\cos\varphi)\,,</math>
+:<math>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).
+where ''φ'' is inclination (or elevation) and ''θ'' is azimuth (right). Notice ''r'' is a constant. Again, the integrals can be separated.
 == See also ==