Ellipsoid Volume CalculatorFree online math calculator — instant results, no sign-up
Calculate an ellipsoid's volume from its three semiaxes.
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About the Ellipsoid Volume Calculator
The Ellipsoid Volume Calculator finds the volume enclosed by an ellipsoid using its three semiaxes, a, b, and c. Each semiaxis is the distance from the center to the surface along one of the ellipsoid’s three principal axes. With the default semiaxes of 3, 4, and 5, the calculator returns a volume of 251.3274122872 cubic units.
- Semiaxis a
- The distance from the center to the surface along the first principal axis, defaulting to 3 and measured in the same length unit as b and c.
- Semiaxis b
- The distance from the center to the surface along the second principal axis, defaulting to 4 and measured in the same length unit as a and c.
- Semiaxis c
- The distance from the center to the surface along the third principal axis, defaulting to 5 and measured in the same length unit as a and b.
Formula and method
V = (4/3)πabc; with a = 3, b = 4, and c = 5: V = (4/3)π(3)(4)(5) = 80π ≈ 251.3274122872.
- VVolume enclosed by the ellipsoid, expressed in cubic units.
- aSemiaxis length along the first principal axis.
- bSemiaxis length along the second principal axis.
- cSemiaxis length along the third principal axis.
- πPi, the mathematical constant approximately equal to 3.141592653589793.
Multiply the three semiaxis lengths, then multiply their product by 4π/3. For the default inputs, abc = 3 × 4 × 5 = 60, so the volume is 80π, displayed as 251.3274122872.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Semiaxis a3
- Semiaxis b4
- Semiaxis c5
Results
- Result251.3274122872
How to use the Ellipsoid Volume Calculator
- 1Identify the three mutually perpendicular principal axes of the ellipsoid and measure each center-to-surface distance, rather than the full end-to-end length.
- 2Convert all three semiaxis measurements to the same length unit.
- 3Enter 3 for Semiaxis a, 4 for Semiaxis b, and 5 for Semiaxis c to reproduce the default example.
- 4Run the calculation to apply V = (4/3)πabc.
- 5Read the Result value of 251.3274122872 as cubic units corresponding to your input length unit.
Reading your result
The Result row reports the ellipsoid’s enclosed volume. For the default inputs, Result = 251.3274122872; if the semiaxes are measured in centimeters, this is 251.3274122872 cm³, and if they are measured in meters, it is 251.3274122872 m³. This output is volume, not surface area.
Limitations
The formula assumes an ideal ellipsoid with three mutually perpendicular principal axes and positive semiaxis lengths. All inputs must use the same length unit; mixed units produce an invalid volume unless converted first. A zero semiaxis describes a degenerate shape with zero volume, while negative lengths have no physical meaning here. For irregular objects, the result is only an ellipsoidal approximation. The decimal output is rounded, and its many digits do not imply equally precise measurements.
Common questions
Should I enter full axis lengths or semiaxis lengths?+
Enter semiaxis lengths: the distances from the center to the surface. If you measured full end-to-end axis lengths, divide each by two before entering it. Entering all three full lengths without halving them makes the calculated volume eight times too large.
Does it matter which semiaxis I call a, b, or c?+
No. The volume depends on the product abc, so rearranging the default lengths 3, 4, and 5 among the inputs leaves the result unchanged at 251.3274122872 cubic units.
Can this calculator also calculate a sphere’s volume?+
Yes. A sphere is an ellipsoid with all three semiaxes equal to its radius. Setting a = b = c = r reduces the formula to V = (4/3)πr³.
Why is the default result a decimal rather than an exact value?+
The default semiaxes give an exact volume of 80π cubic units. Because π has a nonterminating decimal expansion, the displayed value 251.3274122872 is a rounded approximation.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
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Results are estimates for informational purposes and are not financial, medical or legal advice.