Volume of an Ellipsoid Formula

An ellipsoid is a closed quadric surface that is a three-dimensional analogue of an ellipse. The standard equation of an ellipsoid centred at the origin of a Cartesian coordinate system. The spectral theorem can again be used to obtain a standard equation akin to the definition given above.

Ellipsoid Formula

The formula of Ellipsoid is given below:

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

or the formula can also be written as:

\[\large V=\frac{4}{3}\pi\,r_1\,r_2\,r_3\]

a = r1 = Radius of the ellipsoid of axis 1
b = r2 = Radius of the ellipsoid of axis 2
c = r3 = Radius of the ellipsoid of axis 3

Volume of an Ellipsoid Formula Solved Example

Example: The ellipsoid whose radii are given as a = 9 cm, b = 6 cm and c = 3 cm. Find the volume of an ellipsoid.


Radius (a) = 9 cm
Radius (b) = 6 cm
Radius (c) = 3 cm

Using the formula: 

\(\begin{array}{l}V=\frac{4}{3}\pi\,a\,b\,c\end{array} \)
\(\begin{array}{l}V=\frac{4}{3}\times 3.14\times9\times6\times3\end{array} \)
\(\begin{array}{l}V=678.24\,cm^{3}\end{array} \)

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