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.

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:

$\begin{array}{rl}V& =\frac{4}{3}\pi R^3-\frac{4}{3}\pi r^3\\ & =\frac{4}{3}\pi (R^3-r^3)\\ & =\frac{4}{3}\pi (R-r)(R^2+Rr+r^2)\end{array}$

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:

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

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

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

See also

• Spherical pressure vessel
• Ball
• Solid torus
• Bubble
• Sphere