Volume Of Ellipsoid Calculator

What this calculator does

An ellipsoid is a sphere that has been stretched or squashed, a smooth three-dimensional oval. A rugby ball, an egg, a worn pebble, the Earth itself just slightly. Where a sphere is the same distance across in every direction, an ellipsoid can have a different half-width along each of its three perpendicular axes. This works out its volume from those three semi-axes, a, b and c.

Type the three in and you have the space inside.

Using the calculator

1. Type the three semi-axes a, b and c, the half-widths from the centre out to the surface along each axis.
2. Press Calculate.

All three have to be positive. There is no unit setting, so the result is a plain number in cubic units of whatever unit you used.

The formula | volume = (4 × π × a × b × c) ÷ 3

The volume of an ellipsoid is:

volume = (4 × π × a × b × c) ÷ 3

This is the sphere's formula with its single radius split into three. A sphere's volume is (4 × π × radius³) ÷ 3, which is the same as (4 × π × radius × radius × radius) ÷ 3. An ellipsoid simply lets those three radii differ, one for each axis, giving a × b × c in place of radius cubed. So a sphere is just the special case where all three are equal.

Semi-axes, not full widths

The one thing to get right: the three numbers are semi-axes, the half-widths from the centre out to the surface, not the full widths all the way across. It is the same point as using a sphere's radius rather than its diameter. If what you have measured is the full length, width and depth across the whole shape, halve each of them before entering it. Putting in full widths is the usual mistake, and it makes the volume come out eight times too big.

The sphere, and the spheroids

The shape has a family, depending on how many of the three semi-axes match.

• All three equal: a sphere, the same in every direction. See the volume of sphere calculator.
• Two equal, one longer: a prolate spheroid, stretched out like a rugby ball or an American football.
• Two equal, one shorter: an oblate spheroid, flattened like the Earth, which bulges at the equator and flattens at the poles.
• All three different: a general, or triaxial, ellipsoid.

The value of π it uses, and the unitless answer

The tool sets π to 3.141592654, ten significant figures, far finer than any real measurement, so the three semi-axes are what limit the answer, not π. And it does not ask for a unit, reporting a bare number, so work in whatever unit you like and read the result in that unit cubed.

One curiosity worth knowing: while the ellipsoid's volume is this clean and exact, its surface area has no simple exact formula and can only be approximated, the same awkwardness that a stretched circle's perimeter runs into. The volume, though, is no trouble at all.

A worked example | semi-axes 5, 4, 3

Say the three semi-axes are 5, 4 and 3.

1. Multiply the three together: 5 × 4 × 3 = 60.
2. Multiply by 4 and by π: 4 × π × 60 = 753.98.
3. Divide by 3: 753.98 ÷ 3 = 251.33.

So the volume is about 251 cubic units. As a check on the sphere link, set all three to 10 and you get 4,188.79, exactly the volume of a sphere of radius 10.

Where ellipsoids show up

More places than you might expect. Planets and moons are ellipsoids, the Earth an oblate one, and the model used for GPS and mapping is an ellipsoid rather than a true sphere. Rugby and American footballs are prolate spheroids. Eggs and river pebbles are roughly ellipsoidal. And in medical imaging, an organ or a growth is often measured along three perpendicular directions and treated as an ellipsoid to estimate its volume from a scan.

Questions people ask

What is the volume of an ellipsoid with semi-axes 5, 4 and 3?

About 251 cubic units. Multiply the three semi-axes, then by 4 and by π, and divide by 3.

Are the inputs the full widths or the half-widths?

The half-widths, the semi-axes from the centre to the surface. If you have full widths across the shape, halve each first.

What if all three semi-axes are equal?

Then the ellipsoid is a sphere, and the formula becomes the familiar (4 × π × radius³) ÷ 3.

What is a spheroid?

An ellipsoid with two of its three semi-axes equal. Prolate ones are stretched like a rugby ball, oblate ones flattened like the Earth.

What unit is the answer in?

Whatever unit you used, cubed. The tool reports a plain number with no unit attached.

References

A note on where this comes from. The ellipsoid volume (4 × π × a × b × c) ÷ 3 generalises the sphere, recovering (4 × π × radius³) ÷ 3 when the three semi-axes are equal. The Earth is modelled as an oblate spheroid in geodesy, the basis of GPS positioning. The value of π used is the one tabulated by the US National Institute of Standards and Technology. For further reading, see Ellipsoid.

1. The ellipsoid volume formula, (4 × π × a × b × c) ÷ 3, generalising the volume of a sphere to three unequal semi-axes.
2. National Institute of Standards and Technology (NIST), Digital Library of Mathematical Functions, value of π. https://dlmf.nist.gov/