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Volume of an Ellipsoid Formula with Problem Solution & Solved Example


What is the volume of an Ellipsoid?

You must be wondering what is an Ellipsoid? An Ellipsoid is defined as the surface that is obtained with the sphere by deforming it with directional scaling and an affine transformation too. It could also be defined as the quadric sphere that could be defined as zero set of a polynomial of degree having two or three variables. Among quadratic surfaces, this is easy to identify ellipsoid with the following two properties –

Every planar section is an ellipse or it is empty or it can be reduced to a single point like an ellipse. It is bounded by means of an enclosed sufficiently large sphere.

If two axes are of the same length then it is called the ellipsoid of revolution also a spheroid. Further, Ellipsoid is an invariant under rotation around the third axis and there are infinitely plenty of ways for choosing two axes of the same length.

Volume of an Ellipsoid

In case, the third axis is shorter then it is named as the oblate spheroid. If the third axis is longer then it is called as the prolate spheroid. If the three axes are of the same length then the ellipsoid would be a sphere.

The volume of an Ellipsoid Formula

The volume of an Ellipsoid formula in mathematics could be given as –

\[\large V=\frac{4}{3}\pi\,a\,b\,c\]

or the formula can also be written as:

\[\large V=\frac{4}{3}\pi\,r1\,r2\,r3\]

Where,
r1= radius of the ellipsoid 1
r2= radius of the ellipsoid 2
r3= radius of the ellipsoid 3

Further, an ellipsoid has a pairwise perpendicular axis of symmetry that intersects at the center of symmetry that is named as the center of the ellipsoid. The line that delimits on the axis of symmetry by the ellipsoid is called the principal axes or simply the ellipsoid axes. In case, if all three axes have different lengths and they need to be defined uniquely.


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