Area Of A Sector Derivation

Intersection of a sphere and cone emanating from its center

A spherical sector (blueish)

In geometry, a spherical sector,^[1] too known as a spherical cone,^[ii] is a portion of a sphere or of a brawl defined by a conical purlieus 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 operations of the cap. It is the 3-dimensional analogue of the sector of a circumvolve.

Volume [edit]

If the radius of the sphere is denoted by r and the superlative of the cap by h, the volume of the spherical sector is

${\displaystyle V={\frac {2\pi r^{ii}h}{3}}\,.}$

This may also be written as

${\displaystyle V={\frac {2\pi r^{3}}{iii}}(1-\cos \varphi )\,,}$

where φ is one-half the cone bending, 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:

${\displaystyle 5={\frac {rA}{3}}\,.}$

Area [edit]

The curved surface area of the spherical sector (on the surface of the sphere, excluding the cone surface) is

${\displaystyle A=2\pi rh\,.}$

Information technology is besides

${\displaystyle A=\Omega r^{two}}$

where Ω is the solid angle of the spherical sector in steradians, the SI unit of solid bending. One steradian is defined equally the solid bending subtended past a cap area of A = r ².

Derivation [edit]

The volume can exist calculated by integrating the differential volume chemical element

${\displaystyle dV=\rho ^{2}\sin \phi d\rho d\phi d\theta }$

over the volume of the spherical sector,

${\displaystyle V=\int _{0}^{2\pi }\int _{0}^{\varphi }\int _{0}^{r}\rho ^{ii}\sin \phi \,d\rho d\phi d\theta =\int _{0}^{2\pi }d\theta \int _{0}^{\varphi }\sin \phi d\phi \int _{0}^{r}\rho ^{2}d\rho ={\frac {two\pi r^{three}}{3}}(1-\cos \varphi )\,,}$

where the integrals have been separated, because the integrand can be separated into a production of functions each with one dummy variable.

The area can be similarly calculated by integrating the differential spherical area element

${\displaystyle dA=r^{ii}\sin \phi d\phi d\theta }$

over the spherical sector, giving

${\displaystyle A=\int _{0}^{two\pi }\int _{0}^{\varphi }r^{2}\sin \phi d\phi d\theta =r^{2}\int _{0}^{2\pi }d\theta \int _{0}^{\varphi }\sin \phi d\phi =two\pi r^{two}(one-\cos \varphi )\,,}$

where φ is inclination (or summit) and θ is azimuth (right). Find r is a constant. Again, the integrals tin can be separated.

Run into likewise [edit]

• Circular sector — the analogous 2D figure.
• Spherical cap
• Spherical segment
• Spherical wedge

References [edit]

1. ^ Weisstein, Eric Due west. "Spherical sector". MathWorld.
2. ^ Weisstein, Eric W. "Spherical cone". MathWorld.

Area Of A Sector Derivation,

Source: https://en.wikipedia.org/wiki/Spherical_sector

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