# Spherical segment

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Spherical_segment
> Markdown URL: https://mediated.wiki/source/Spherical_segment.md
> Source: https://en.wikipedia.org/wiki/Spherical_segment
> Source revision: 1288178551
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

Region between parallel planes intersecting a sphere

This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations. (May 2024) (Learn how and when to remove this message)

A spherical segment

Pair of parallel planes intersecting a sphere forming a spherical segment (i.e., a spherical frustum)

Terminology for spherical segments.

In [geometry](/source/Geometry), a **spherical segment** is the [solid](/source/Solid_(geometry)) defined by cutting a [sphere](/source/Sphere) or a [ball](/source/Ball_(mathematics)) with a pair of [parallel planes](/source/Parallel_(geometry)). It can be thought of as a [spherical cap](/source/Spherical_cap) with the top truncated, and so it corresponds to a **spherical [frustum](/source/Frustum)**.

The [surface](/source/Surface_(mathematics)) of the *spherical segment* (excluding the bases) is called **spherical zone**.

Geometric parameters for spherical segment.

If the [radius](/source/Radius) of the sphere is called R, the radii of the spherical segment bases are a and b, and the height of the segment (the distance from one parallel plane to the other) called h, then the [volume](/source/Volume) of the spherical segment is

- V = π 6 h ( 3 a 2 + 3 b 2 + h 2 ) . {\displaystyle V={\frac {\pi }{6}}h\left(3a^{2}+3b^{2}+h^{2}\right).}

For the special case of the top plane being tangent to the sphere, we have b = 0 {\displaystyle b=0} and the solid reduces to a [spherical cap](/source/Spherical_cap).[1]

The equation above for volume of the spherical segment can be arranged to

- V = [ π a 2 ( h 2 ) ] + [ π b 2 ( h 2 ) ] + [ 4 3 π ( h 2 ) 3 ] {\displaystyle V={\biggl [}\pi a^{2}\left({\frac {h}{2}}{\biggr )}\right]+{\biggl [}\pi b^{2}\left({\frac {h}{2}}{\biggr )}\right]+{\biggl [}{\frac {4}{3}}\pi \left({\frac {h}{2}}\right)^{3}{\biggr ]}}

Thus, the segment volume equals the sum of three volumes: two right circular cylinders one of radius a and the second of radius b (both of height h / 2 {\displaystyle h/2} ) and a sphere of radius h / 2 {\displaystyle h/2} .

The curved [surface area](/source/Surface_area) of the spherical zone—which excludes the top and bottom bases—is given by

- A = 2 π R h . {\displaystyle A=2\pi Rh.}

Thus the surface area of the segment depends only on the distance between the cutting planes, and not their absolute heights.

## See also

- [Spherical cap](/source/Spherical_cap)

- [Spherical wedge](/source/Spherical_wedge)

- [Spherical sector](/source/Spherical_sector)

## References

1. **[^](#cite_ref-1)** Kern, Willis; Bland, James (1938). [*Solid Mensuration with Proofs*](https://archive.org/details/in.ernet.dli.2015.205959) (Second ed.). New York: John Wiley & Sons, Inc. pp. 97–103. Retrieved 16 May 2024.

- Kern, William F.; Bland, James R. (1938). [*Solid Mensuration with Proofs*](https://archive.org/details/in.ernet.dli.2015.205959). p. [95](https://archive.org/details/in.ernet.dli.2015.205959/page/n103)–97.

## External links

Wikimedia Commons has media related to [Spherical segments](https://commons.wikimedia.org/wiki/Category:Spherical_segments).

- [Weisstein, Eric W.](/source/Eric_W._Weisstein) ["Spherical segment"](https://mathworld.wolfram.com/SphericalSegment.html). *[MathWorld](/source/MathWorld)*.

- [Weisstein, Eric W.](/source/Eric_W._Weisstein) ["Spherical zone"](https://mathworld.wolfram.com/Zone.html). *[MathWorld](/source/MathWorld)*.

- [Summary of spherical formulas](http://mathforum.org/dr.math/faq/formulas/faq.sphere.html#spheresegment)

This geometry-related article is a stub. You can help Wikipedia by adding missing information.

- [v](https://en.wikipedia.org/wiki/Template:Geometry-stub)
- [t](/source/Template_talk%3AGeometry-stub)
- [e](https://en.wikipedia.org/wiki/Special:EditPage/Template:Geometry-stub)

---
Adapted from the Wikipedia article [Spherical segment](https://en.wikipedia.org/wiki/Spherical_segment) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Spherical_segment?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.