Volume of an Ellipsoid Calculator
Calculate the volume and approximate surface area of an ellipsoid from its three semi-axes. See also Volume of Sphere Calculator and Area of Ellipse Calculator.
How to Calculate the Volume of an Ellipsoid
An ellipsoid is a 3D shape where every cross-section is an ellipse (or circle). It has three semi-axes — a, b, and c — measured along three perpendicular directions. To find the volume, multiply all three semi-axes together, multiply by π, then multiply by 4/3. When all three axes are equal, the ellipsoid becomes a sphere. The surface area of a general ellipsoid has no simple closed-form formula, so this calculator uses the Knud Thomsen approximation.
Ellipsoid Volume Formula
V = (4/3) × π × a × b × c
Approximate Surface Area (Knud Thomsen):
SA ≈ 4π × ((apbp + apcp + bpcp) / 3)1/p
where p ≈ 1.6075
Example
Find the volume of an ellipsoid with a=6, b=4, c=3:
V = (4/3) × π × a × b × c
V = (4/3) × π × 6 × 4 × 3
V = (4/3) × π × 72
V ≈ 301.5929 cubic units
Ellipsoid Volume Reference Table
| a | b | c | Volume |
|---|---|---|---|
| 1 | 1 | 1 | 4.1888 |
| 2 | 1 | 1 | 8.3776 |
| 2 | 2 | 1 | 16.7552 |
| 3 | 2 | 1 | 25.1327 |
| 3 | 2 | 2 | 50.2655 |
| 4 | 3 | 2 | 100.5310 |
| 5 | 3 | 2 | 125.6637 |
| 5 | 4 | 3 | 251.3274 |
| 6 | 4 | 3 | 301.5929 |
| 6 | 5 | 4 | 502.6548 |
| 8 | 5 | 3 | 502.6548 |
| 8 | 6 | 4 | 804.2477 |
| 10 | 6 | 4 | 1005.3096 |
| 10 | 8 | 5 | 1675.5161 |
| 10 | 10 | 10 | 4188.7902 |
Frequently Asked Questions
What is an ellipsoid?
An ellipsoid is a 3D surface where every cross-section through the center is an ellipse. It is defined by three semi-axes (a, b, c) along three perpendicular directions. When all three are equal, it becomes a sphere.
What is the Knud Thomsen approximation?
Unlike the volume, the surface area of a general ellipsoid cannot be expressed with a simple formula. The Knud Thomsen approximation provides an estimate with a relative error of at most about 1.061%. It uses the parameter p ≈ 1.6075.
What is the difference between an ellipsoid and a spheroid?
A spheroid is a special ellipsoid where two of the three semi-axes are equal. An oblate spheroid (like Earth) has a = b > c. A prolate spheroid (like a rugby ball) has a > b = c. A general ellipsoid has all three axes different.
Does the order of a, b, c matter for volume?
No. Since multiplication is commutative, V = (4/3)πabc gives the same result regardless of which axis you label a, b, or c.
What are real-world examples of ellipsoids?
Earth is approximately an oblate spheroid. Eggs, watermelons, and rugby balls are roughly ellipsoidal. In physics, atomic orbitals and gravitational fields often have ellipsoidal shapes.