Ellipsoid Volume Calculator
An ellipsoid is a three-dimensional generalization of an ellipse, defined by three semi-axes a, b, and c extending from the center. The volume of an ellipsoid is simply V = (4/3) * π * a * b * c. The surface area is more complex and is approximated using Knud Thomsen's formula: A ≈ 4π * ((a^p*b^p + a^p*c^p + b^p*c^p)/3)^(1/p) where p ≈ 1.6075. This calculator computes both the volume and surface area of an ellipsoid from the three semi-axes.
Ellipsoid volume formulas
Volume = (4/3) * π * a * b * c
Surface Area (Thomsen) = 4π * ((a^p*b^p + a^p*c^p + b^p*c^p)/3)^(1/p)
where p ≈ 1.6075
Reference values
| Semi-axis a | Semi-axis b | Semi-axis c | Volume |
|---|---|---|---|
| 1 cm | 1 cm | 1 cm | 4.19 cm³ |
| 3 cm | 2 cm | 2 cm | 50.27 cm³ |
| 5 cm | 4 cm | 3 cm | 251.33 cm³ |
| 1 m | 1 m | 0.5 m | 2.09 m³ |
Ellipsoid volume calculator: frequently asked questions
What is an ellipsoid?
An ellipsoid is a three-dimensional shape that is like a stretched or squashed sphere. It is defined by three semi-axes a, b, and c extending from the center.
What is the volume formula?
The volume of an ellipsoid is V = (4/3) * π * a * b * c, where a, b, and c are the three semi-axes.
What is the surface area formula?
The exact surface area is complex, but approximated by Knud Thomsen: A ≈ 4π * ((a^p*b^p + a^p*c^p + b^p*c^p)/3)^(1/p) where p ≈ 1.6075.
What is the relationship to a sphere?
A sphere is a special case of an ellipsoid where a = b = c. If two axes are equal, it is called a spheroid (prolate or oblate).
Where are ellipsoids found?
Ellipsoids appear in planetary shapes (Earth), atomic nuclei, cells, and various engineering applications.
Official sources
- Khan Academy: Volume and surface area of solids.
- NIST: National Institute of Standards and Technology.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.