Spherical shell

In geometry, a spherical shell (a ball shell) is a generalization of an annulus to three dimensions. It is the region of a ball between two concentric spheres of differing radii.^[1]

Volume

The volume of a spherical shell is the difference between the enclosed volume of the outer sphere and the enclosed volume of the inner sphere:

\begin{matrix} V & = \frac{4}{3} π R^{3} - \frac{4}{3} π r^{3} \\ = \frac{4}{3} π (R^{3} - r^{3}) \\ = \frac{4}{3} π (R - r) (R^{2} + R r + r^{2}) \end{matrix}

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

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 \approx 4 π r^{2} t,

when $t$ is very small compared to $r$ ( $t ≪ r$ ).

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

^ Page Module:Citation/CS1/styles.css has no content.Weisstein, Eric W. "Spherical Shell". mathworld.wolfram.com. Wolfram Research, Inc. Archived from the original on 2 August 2016. Retrieved 7 January 2017.
^ Page Module:Citation/CS1/styles.css has no content.Znamenski, Andrey Varlamov; Lev Aslamazov (2012). A.A. Abrikosov Jr. (ed.). The wonders of physics. Translated by A.A. Abrikosov Jr.; J. Vydryg; D. Znamenski (3rd ed.). Singapore: World Scientific. p. 78. ISBN 978-981-4374-15-6.