Volume of an Ellipsoid Formula: Equation & Types

Volume of an Ellipsoid can be defined as the amount of space enclosed by it. We all know that an ellipsoid is a closed three dimensional geometric figure which gets its name from another closed geometrical figure, i.e, ellipse. An ellipsoid has three axes of rotational symmetry which are useful in finding the volume of a given ellipsoid. On the basis of the length of these axes, we can also categorize an ellipsoid into two types. Let’s have a closer look at the topic and discuss some solved examples and important questions.

Key Terms: Ellipse, Ellipsoid, Volume, Symmetry, Volume of an Ellipsoid Formula, circles, centre of ellipsoid.

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Ellipsoid and its Types

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Ellipsoid may be defined as a closed three dimensional figure with ellipses or circles as its cross sectional planes. An Ellipsoid can be considered analogous to an ellipse due to the reason that any plane that passes through or cuts the ellipsoid will result in the formation of an ellipse. As mentioned above, an ellipsoid has three axes of rotational symmetry which are perpendicular to each other. The point where these three perpendicular axes meet is considered as the centre of ellipsoid. 

For instance, let us consider an ellipsoid with a, b and c as three axes of rotational symmetry. On the basis of varying degree of length of these three axes, we classify an ellipsoid into following two types:

1. Oblate Ellipsoid: If the length of two axes of rotational symmetry of an ellipsoid are equal and one of them is also greater than the third axes of rotational symmetry, then it is known as Oblate Ellipsoid. Mathematically, a=b and a>c is the condition that needs to be satisfied in order to get an oblate ellipsoid.
2. Prolate Ellipsoid: If the length of two axes of rotational symmetry of an ellipsoid are equal and one of them is also smaller than the third axes of rotational symmetry, then it is known as Prolate Ellipsoid. Mathematically, a=b and c>a is the condition that needs to be satisfied in order to get a prolate ellipsoid.

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Equation of Ellipsoid 

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The standard equation of an ellipsoid with a, b, c as lengths of semi-axes of ellipsoid can be represented by

(x²/a²) + (y²/b²) + (z²/c²) = 1

Here, a, b and c are not equal in lengths.

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Volume of an Ellipsoid Formula

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An ellipsoid is referred to as a closed quadric surface which is a three-dimensional analogue of an ellipse. 

Let us consider an ellipsoid with centre O and a, b, c as lengths of semi axes, then the volume of the given ellipsoid can be calculated by using the formula

Volume of Ellipsoid = 4/3 x (πabc) 

Or

Volume of Ellipsoid = 4/3 x r₁ x r₂ x r₃

Where,

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

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Solved Examples

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

Solution: Given,

Radius (a) = 9 cm

Radius (b) = 6 cm

Radius (c) = 3 cm

Using the formula: 

Volume of Ellipsoid = 4/3 x (πabc) 

Volume of Ellipsoid = 4/3 x r₁ x r₂ x r₃

V = 678.24 cm³

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Things to Remember

• An Ellipsoid is a closed quadratic figure with three lengths of semi axes.
• If the lengths of all three axes of rotational symmetry of an ellipsoid are equal, i.e, if a=b=c, then the resulting ellipsoid is a sphere.
• The longest amongst the three semi axes is known as major axes and the shorter two are known as minor axes.
• Volume of Prolate Ellipsoid can be calculated by using the formula, V = 4/3 x (π.a.b.b) = 4/3 (πab²)
• Volume of Oblate Ellipsoid can be calculated by using the formula, V = 4/3 x (π.a.a.b) = 4/3 (πa²b)