# Spherical shell

spherical shell, right: two halves

In geometry, a spherical 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:

${\displaystyle V={\frac {4}{3}}\pi R^{3}-{\frac {4}{3}}\pi r^{3}}$
${\displaystyle V={\frac {4}{3}}\pi (R^{3}-r^{3})}$
${\displaystyle V={\frac {4}{3}}\pi (R-r)(R^{2}+Rr+r^{2})}$

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]

${\displaystyle V\approx 4\pi r^{2}t,}$

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

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

2. ^ Znamenski, Andrey Varlamov, Lev Aslamazov; scientific editor, A.A. Abrikosov, Jr. ; translators, A.A. Abrikosov, Jr., J. Vydryg, & D. (2012). The wonders of physics (3rd ed.). Singapore: World Scientific. p. 78. ISBN 978-9814374156. Archived from the original on 20 December 2017. Retrieved 7 January 2017. {{cite book}}: |first1= has generic name (help)