volume of ellipsoid

Let us determine the volume of the ellipsoidMathworldPlanetmathPlanetmath

x2a2+y2b2+z2c2= 1.

Suppose  -axa. When we cut the ellipsoid with a plane parallelMathworldPlanetmathPlanetmath to the yz-plane, that is, let x be , we get the ellipsePlanetmathPlanetmath

y2b2+z2c2= 1-x2a2,


y2b2(1-x2a2)+z2c2(1-x2a2)= 1,

with the semiaxes


The area of this ellipse is πb1c1 (see area of plane region), and thus we have the function


expressing the area cut of the ellipsoid by parallel planes. By the volume formula of the parent entry (http://planetmath.org/VolumeAsIntegral) we can calculate the volume of the ellipsoid as


The special case  a=b=c=r  of a sphere is the well-known expression 43πr3.

Title volume of ellipsoid
Canonical name VolumeOfEllipsoid
Date of creation 2013-03-22 17:20:41
Last modified on 2013-03-22 17:20:41
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Result
Classification msc 51M25
Synonym ellipsoid volume
Related topic Ellipsoid
Related topic SubstitutionNotation
Related topic SqueezingMathbbRn