spherical shell

{{Short description|Three-dimensional geometric shape}}

File:Kugelschale.svg

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 {{mvar|r}} is the radius of the inner sphere and {{mvar|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 {{mvar|t}} of the shell:

: V \approx 4 \pi r^2 t,

when {{mvar|t}} is very small compared to {{mvar|r}} (t \ll r).

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

See also

References

{{reflist|refs=

{{cite web |author=Weisstein, Eric W. |title=Spherical Shell|url=http://mathworld.wolfram.com/SphericalShell.html|website=mathworld.wolfram.com|publisher=Wolfram Research, Inc.|access-date=7 January 2017|language=en|url-status=live|archive-url=https://web.archive.org/web/20160802000613/http://mathworld.wolfram.com/SphericalShell.html|archive-date=2 August 2016}}

{{cite book|last1=Znamenski|first1=Andrey Varlamov |author2=Lev Aslamazov |editor=A.A. Abrikosov Jr. |translator=A.A. Abrikosov Jr. |translator2=J. Vydryg |translator3=D. Znamenski |title=The wonders of physics|date=2012|publisher=World Scientific|location=Singapore|isbn=978-981-4374-15-6|page=78|edition=3rd |url=https://books.google.com/books?id=xh48DQAAQBAJ&pg=PA78}}

}}

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Category:Elementary geometry

Category:Geometric shapes

Category:Spherical geometry

Category:Euclidean solid geometry