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 |
When Do You Need This Calculation?
Ellipsoid volume calculations arise in science, engineering, and everyday life:
- Egg volume: Food scientists and biologists calculate egg volumes for quality grading, incubation studies, and nutritional analysis.
- Rugby/American football: Sports equipment manufacturers calculate ball volumes for pressure testing and material requirements.
- Planetary science: Astronomers model planets and moons as oblate or prolate spheroids to calculate mass and density.
- Medical imaging: Doctors estimate organ or tumor volumes from CT/MRI scans by fitting ellipsoid models.
- Grape/fruit sizing: Agricultural scientists approximate fruit volumes as ellipsoids for yield estimation.
Solved Examples
Example 1: Chicken Egg Volume
A large chicken egg can be approximated as an ellipsoid with semi-axes a=3 cm, b=2.5 cm, c=2.5 cm. Find its volume.
V = (4/3) × π × 3 × 2.5 × 2.5
V = (4/3) × π × 18.75
V ≈ 78.54 cm³ ≈ 78.5 mL
Example 2: Rugby Ball Volume
A rugby ball is approximately a prolate spheroid with semi-axes a=14 cm, b=9 cm, c=9 cm. Calculate its volume.
V = (4/3) × π × 14 × 9 × 9
V = (4/3) × π × 1134
V ≈ 4,750.09 cm³ ≈ 4.75 liters
Example 3: Earth as an Oblate Spheroid
Earth can be modeled as an oblate spheroid with equatorial radius 6,378 km and polar radius 6,357 km. Estimate its volume.
V = (4/3) × π × 6378 × 6378 × 6357
V = (4/3) × π × 2.586 × 10¹¹
V ≈ 1.083 × 10¹² km³
Common Mistakes
Using full axes instead of semi-axes: The formula uses semi-axes (half the full axis length). If an egg is 6 cm long, the semi-axis is 3 cm, not 6 cm.
Adding instead of multiplying axes: The formula multiplies a×b×c, not adds. V = (4/3)π(a×b×c), not (4/3)π(a+b+c).
Using sphere formula for non-spheres: If all three axes differ, you must use the ellipsoid formula. The sphere formula (4/3)πr³ only works when a=b=c.
Assuming exact surface area: The surface area formula is an approximation (Knud Thomsen). Only the volume formula is exact.
Key Takeaways
- Ellipsoid volume = (4/3)πabc — same structure as sphere but with three different semi-axes.
- When a=b=c, the ellipsoid becomes a sphere. When two axes are equal, it's a spheroid.
- Use semi-axes (half of full dimensions), not the full diameter/axis measurements.
- The surface area has no exact closed-form — the Knud Thomsen approximation is within ~1% error.
- Common ellipsoids: eggs, rugby balls, planets, grapes, and medical organ models.
- The order of a, b, c doesn't matter for volume since multiplication is commutative.
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.