Mironenko reflecting function

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A spherical sector

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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.

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:

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

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If the radius of the base of the cap is denoted by a, the surface area of the spherical sector is

$A=\pi r(2h+a).$

Derivation

$V=\underset{0}{\overset{2\pi }{\int }}\underset{0}{\overset{\varphi }{\int }}\underset{0}{\overset{r}{\int }}{r}^{2}sin\varphi \mathrm{d}r\mathrm{d}\varphi \mathrm{d}\theta =\frac{2\pi r^3}{3}\underset{0}{\overset{\varphi }{\int }}sin\varphi \mathrm{d}\varphi =\frac{2\pi r^3}{3}(1-\cos \varphi )$

Where $\varphi$ for inclination (or elevation) and $\theta$ for azimuth (right).

See also

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

External links

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• Summary of spherical formulas

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