# Spherical shell

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Three-dimensional geometric shape

spherical shell, right: two halves

In [geometry](/source/Geometry), a **spherical shell** (a **[ball shell](/source/Ball_(mathematics)#Regions)**) is a generalization of an [annulus](/source/Annulus_(mathematics)) to three dimensions. It is the region of a [ball](/source/Ball_(mathematics)) between two [concentric](/source/Concentric) [spheres](/source/Sphere) of differing radii.[1]

## Volume

The [volume](/source/Volume) of a spherical shell is the difference between the enclosed volume of the outer sphere and the [enclosed volume of the inner sphere](/source/Sphere#Enclosed_volume):

- V = 4 3 π R 3 − 4 3 π r 3 = 4 3 π ( R 3 − r 3 ) = 4 3 π ( R − r ) ( R 2 + R r + r 2 ) {\displaystyle {\begin{aligned}V&={\tfrac {4}{3}}\pi R^{3}-{\tfrac {4}{3}}\pi r^{3}\\[3mu]&={\tfrac {4}{3}}\pi {\bigl (}R^{3}-r^{3}{\bigr )}\\[3mu]&={\tfrac {4}{3}}\pi (R-r){\bigl (}R^{2}+Rr+r^{2}{\bigr )}\end{aligned}}}

where r is the radius of the inner sphere and R is the radius of the outer sphere.

## Approximation

An approximation for the volume of a thin spherical shell is the surface area of the inner sphere multiplied by the thickness t of the shell:[2]

- V ≈ 4 π r 2 t , {\displaystyle V\approx 4\pi r^{2}t,}

when t is very small compared to r ( t ≪ r {\displaystyle t\ll r} ).

The total surface area of the spherical shell is 4 π r 2 {\displaystyle 4\pi r^{2}} .

## See also

- [Spherical pressure vessel](/source/Pressure_vessel#Spherical_vessel)

- [Ball](/source/Ball_(mathematics))

- [Solid torus](/source/Solid_torus)

- [Bubble](/source/Bubble_(physics))

- [Sphere](/source/Sphere)

- [Focaloid](/source/Focaloid)

## References

1. **[^](#cite_ref-Wolfram_1-0)** Weisstein, Eric W. ["Spherical Shell"](http://mathworld.wolfram.com/SphericalShell.html). *mathworld.wolfram.com*. Wolfram Research, Inc. [Archived](https://web.archive.org/web/20160802000613/http://mathworld.wolfram.com/SphericalShell.html) from the original on 2 August 2016. Retrieved 7 January 2017.

1. **[^](#cite_ref-wonders_2-0)** Znamenski, Andrey Varlamov; Lev Aslamazov (2012). A.A. Abrikosov Jr. (ed.). [*The wonders of physics*](https://books.google.com/books?id=xh48DQAAQBAJ&pg=PA78). Translated by A.A. Abrikosov Jr.; J. Vydryg; D. Znamenski (3rd ed.). Singapore: World Scientific. p. 78. [ISBN](/source/ISBN_(identifier)) [978-981-4374-15-6](https://en.wikipedia.org/wiki/Special:BookSources/978-981-4374-15-6).

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