geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a spherical shell is a generalization of an

annulus Annulus (or anulus) or annular indicates a ring- or donut-shaped area or structure. It may refer to: Human anatomy * ''Anulus fibrosus disci intervertebralis'', spinal structure * Annulus of Zinn, a.k.a. annular tendon or ''anulus tendineus com ...

to three dimensions. It is the region of a

ball A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used f ...

between two concentric

sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

s of differing radii.

Volume

The

volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...

of a spherical shell is the difference between the enclosed volume of the outer sphere and the enclosed volume of the inner sphere: :

V=\frac\pi R^3- \frac\pi r^3

V=\frac\pi (R^3-r^3)

V=\frac\pi (R-r)(R^2+Rr+r^2)

where is the radius of the inner sphere and 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 of the shell: :

V \approx 4 \pi r^2 t,

when is very small compared to (

t \ll r

). The total surface area of the spherical shell is

4 \pi r^2

References

{{DEFAULTSORT:Spherical Shell Elementary geometry Geometric shapes Spherical geometry

Volume

Approximation

See also

References