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Theorem — what does that mean?
A sphere stretched along three axes: four thirds of pi times the three semi-axes.
In Swedish: Ellipsoidens volym
V
A sphere stretched along three axes: four thirds of pi times the three semi-axes.
V≈100.531cm³
V equals ((4 times π) divided by 3) times (4 times (3 times 2)), is approximately (12.5664 divided by 3) times (4 times 6), is approximately 4.1888 times 24, is approximately 100.531 cm³
You are given
Round to 1 decimal.
Pick the unit your lengths are in. Areas are then labelled in square units and volumes in cubic units; nothing is converted.
a, b and c are the three semi-axes: half the length of the solid along each of its three axes. With a = b = c = r the formula is the sphere’s, 4πr³ / 3, and the ellipsoid is what that becomes when the sphere is stretched by different amounts in three directions.
Semi-axes of 4, 3 and 2 cm: V = 4π · 4 · 3 · 2 / 3 = 32π ≈ 100.5 cm³. A sphere with the same volume would have a radius of ∛24 ≈ 2.9 cm, the geometric mean of the three.
Stretching a solid by a factor k along one axis multiplies its volume by k, whichever shape it is. A unit sphere stretched by a, b and c along the three axes has volume 4π/3 · abc, which is the formula. That is also why the formula holds for any ellipsoid, however lopsided, as long as the three axes are at right angles.
An egg is close to an ellipsoid with semi-axes of about 3, 2.2 and 2.2 cm: 4π · 3 · 2.2 · 2.2 / 3 ≈ 61 cm³, about 60 g at the density of water. The approximation is rough because an egg is blunter at one end than the other, which an ellipsoid is not.
What is a spheroid?
An ellipsoid with two equal semi-axes, as the Earth is: a = b at the equator and c a little shorter
at the poles.
Is there a surface area formula?
Not a closed one for the general ellipsoid; spheroids have one with an arcsine or an inverse
hyperbolic sine in it. That is why only the volume is in the collection.
How does the ellipse relate to it?
A cut through the centre is an ellipse with area πab, and the ellipsoid’s
volume is the ellipse’s area times 4c / 3.
| Solve for | Rearranged | With the numbers | Comes to |
|---|---|---|---|
| a | 3V4πbc | 3·100.54π·3·2 | 4 cm |
| V | 4π3abc | 4π3·4·3·2 | 100.5 cm³ |
Last reviewed October 20, 2026 ·How pages are reviewed
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