Spherical shell

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{align}

V &= \tfrac43\pi R^3 - \tfrac43\pi r^3 \\[3mu] &= \tfrac43\pi \bigl(R^3 - r^3\bigr) \\[3mu] &= \tfrac43\pi (R-r)\bigl(R^2 + Rr + r^2\bigr) \end{align}$$ 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 $$4 \pi r^2$$.