Spherical shell

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title: "Spherical shell" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["elementary-geometry", "geometric-shapes", "spherical-geometry", "euclidean-solid-geometry"] description: "Three-dimensional geometric shape" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Spherical_shell" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Three-dimensional geometric shape ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/0/07/Kugelschale.svg" caption="spherical shell, right: two halves"] ::

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.

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.

References

References

1. Weisstein, Eric W.. "Spherical Shell". Wolfram Research, Inc..
2. (2012). "The wonders of physics". World Scientific.

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::