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# volume of ellipsoid

Let us determine the volume of the ellipsoid

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}\;=\;1.$ |

Suppose $-a\leqq x\leqq a$. When we cut the ellipsoid with a plane parallel to the $yz$-plane, that is, let $x$ be constant, we get the ellipse

$\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}\;=\;1\!-\!\frac{x^{2}}{a^{2}},$ |

i.e.

$\frac{y^{2}}{b^{2}\left(1\!-\!\frac{x^{2}}{a^{2}}\right)}+\frac{z^{2}}{c^{2}% \left(1\!-\!\frac{x^{2}}{a^{2}}\right)}\;=\;1,$ |

with the semiaxes

$b_{1}:=b\sqrt{1\!-\!\frac{x^{2}}{a^{2}}},\quad c_{1}\;:=\;c\sqrt{1\!-\!\frac{x% ^{2}}{a^{2}}}.$ |

The area of this ellipse is $\pi b_{1}c_{1}$ (see area of plane region), and thus we have the function

$A(x)\;:=\;\pi bc\left(1-\frac{x^{2}}{a^{2}}\right)$ |

expressing the area cut of the ellipsoid by parallel planes. By the volume formula of the parent entry we can calculate the volume of the ellipsoid as

$V\;=\;\int_{{-a}}^{a}\!A(x)\,dx=\pi bc\int_{{-a}}^{a}\!\left(1\!-\!\frac{x^{2}% }{a^{2}}\right)\,dx\;=\;\pi bc\!\operatornamewithlimits{\Big/}_{{\!\!\!x\,=-a}% }^{{\,\quad a}}\left(x-\frac{x^{3}}{3a^{2}}\right)\;=\;\frac{4}{3}\pi abc.$ |

The special case $a=b=c=r$ of a sphere is the well-known expression $\frac{4}{3}\pi r^{3}.$

Related:

Ellipsoid, SubstitutionNotation, SqueezingMathbbRn

Synonym:

ellipsoid volume

Type of Math Object:

Result

Major Section:

Reference

Parent:

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## Comments

## Ellipsoid Volume

Why do some people (particularly from the medical field) use the following equation to calculate ellipsoid volume?

volume = length Ã— width Ã— thickness Ã— Ï€/6

## Re: Ellipsoid Volume

Volume = Length X width X thickness X pi/6

rather than using Length X width X thickness X 4/3 X pi

## Re: Ellipsoid Volume

> Volume = Length X width X thickness X pi/6

> rather than using Length X width X thickness X 4/3 X pi

That's easy. The second formula would be valid if you were using the half-length, half-width and half-thickness (like the radius of the sphere). If you use the full lengths (like the diameter) then you must divide the answer by 2x2x2=8, so 4/3 becomes 1/6.

## Re: Ellipsoid Volume

Hi ansonkoehler,

See the last Property in http://planetmath.org/encyclopedia/Similar2.html

Regards,

Jussi