Spherical sector#Volume

If the radius of the sphere is denoted by {{mvar|r}} and the height of the cap by {{mvar|h}}, the volume of the spherical sector is

$$V\; =\; \backslash frac\{2\backslash pi\; r^2\; h\}\{3\}\backslash ,.$$

This may also be written as

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

where {{mvar|φ}} is half the cone aperture angle, i.e., {{mvar|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 {{mvar|φ}} approaching 180 degrees, which then describes a complete sphere.

The height, {{mvar|h}} is given by

$$h\; =\; r\; (1-\backslash cos\backslash varphi)\backslash ,.$$

The volume {{mvar|V}} of the sector is related to the area {{mvar|A}} of the cap by:

$$V\; =\; \backslash frac\{rA\}\{3\}\backslash ,.$$

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

$$A\; =\; 2\backslash pi\; rh\backslash ,.$$

It is also

$$A\; =\; \backslash Omega\; r^2$$

where {{math|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 {{math|1=A = r²}}.

{{further|double integral|triple integral}}

The volume can be calculated by integrating the differential volume element

$$dV\; =\; \backslash rho^2\; \backslash sin\; \backslash phi\; \backslash ,\; d\backslash rho\; \backslash ,\; d\backslash phi\; \backslash ,\; d\backslash theta$$

over the volume of the spherical sector,

$$V\; =\; \backslash int\_0^\{2\backslash pi\}\; \backslash int\_0^\backslash varphi\backslash int\_0^r\backslash rho^2\backslash sin\backslash phi\; \backslash ,\; d\backslash rho\; \backslash ,\; d\backslash phi\; \backslash ,\; d\backslash theta\; =\; \backslash int\_0^\{2\backslash pi\}\; d\backslash theta\; \backslash int\_0^\backslash varphi\; \backslash sin\backslash phi\; \backslash ,\; d\backslash phi\; \backslash int\_0^r\; \backslash rho^2\; d\backslash rho\; =\; \backslash frac\{2\backslash pi\; r^3\}\{3\}\; (1-\backslash cos\backslash varphi)\; \backslash ,\; ,$$

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

$$dA\; =\; r^2\; \backslash sin\backslash phi\; \backslash ,\; d\backslash phi\; \backslash ,\; d\backslash theta$$

over the spherical sector, giving

$$A\; =\; \backslash int\_0^\{2\backslash pi\}\; \backslash int\_0^\backslash varphi\; r^2\; \backslash sin\backslash phi\; \backslash ,\; d\backslash phi\; \backslash ,\; d\backslash theta\; =\; r^2\; \backslash int\_0^\{2\backslash pi\}\; d\backslash theta\; \backslash int\_0^\backslash varphi\; \backslash sin\backslash phi\; \backslash ,\; d\backslash phi\; =\; 2\backslash pi\; r^2(1-\backslash cos\backslash varphi)\; \backslash ,\; ,$$

where {{mvar|φ}} is inclination (or elevation) and {{mvar|θ}} is azimuth (right). Notice {{math|r}} is a constant. Again, the integrals can be separated.

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

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Category:Spherical geometry

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