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 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:

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

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]

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

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

In popular culture

A Dyson sphere encloses a fictitious spherical shell around a star, as first described by author Olaf Stapledon.