How to use this ellipsoid calculator
- Measure each full principal diameter and divide it by two before entry; this page requires semi-axes.
- Enter semi-axes a, b and c in the same length unit. Their order does not change the result.
- Select Calculate. Volume is exact for the supplied semi-axes; surface area is clearly marked as approximate.
- Editing an axis invalidates the answer and Copy. Clear empties all three axes.
Volume and approximate surface area
An ellipsoid satisfies x²/a² + y²/b² + z²/c² = 1, where a, b and c are semi-axes rather than full diameters.
Its volume has the exact formula V = (4/3)πabc. A general triaxial ellipsoid’s surface area is more complex, so this tool uses a practical approximation.
Knud Thomsen’s expression uses p = 1.6075. It is exact when all axes are equal (a sphere); its commonly reported worst-case relative error is about 1.061%, so it should not be treated as an exact survey or manufacturing value.
V = (4/3)πabc · S ≈ 4π[((ab)ᵖ + (ac)ᵖ + (bc)ᵖ)/3]¹ᐟᵖ, p = 1.6075
Worked examples
Semi-axes 3, 2 and 1 cm
Volume is 8π ≈ 25.13 cm³. Thomsen’s formula gives an approximate surface area of 48.97 cm².
Three equal semi-axes of 5 m
The ellipsoid is a sphere. Volume is 500π/3 ≈ 523.6 m³ and the approximation becomes the exact sphere area 100π ≈ 314.16 m².
Limits and input notes
- Enter semi-axes, not full widths. A measured diameter must be halved first.
- All inputs must be positive decimals using one consistent unit.
- The surface result is an approximation with about 1.061% worst-case relative error, not a guaranteed tolerance for a physical object.
- Real irregular objects, wall thickness and cut-outs are outside the mathematical ellipsoid model.
Frequently asked questions
Why is surface area approximate?
The general three-axis surface area involves elliptic integrals; Thomsen’s formula provides a compact, accurate estimate.
Is volume also approximate?
No. V = (4/3)πabc is exact for the ellipsoid defined by the entered semi-axes.
What if two axes are equal?
The shape is a spheroid. This calculator still applies the same volume and Thomsen surface estimate.
Does axis order matter?
No. Both formulas are symmetric in a, b and c.