Ellipsoid Volume Calculator
Volume from three semi axes with sphere match.
What it does
Ellipsoids stretch spheres along three axes: rugby balls, watermelons, and planet bulges all answer to the same product of semi axes. One formula covers spheres, spheroids, and full triaxials alike.
The tool above takes three semi axes in any order. It returns volume with the equal sphere diameter beside it, in cubic inches or centimeters.
The defaults (3, 4, 5) hold 251.33 cu in, matching a 7.83 in sphere. The metric mirror (7.62, 10.16, 12.7 cm) gives 4,118.52 cu cm with a 19.89 cm sphere.
The line follows OpenStax Calculus triple integrals with the first octant symmetry proof ending at 4/3 pi a b c. MIT single variable calculus derives the same product by the disk method.
How to use
- Pick imperial (in) or metric (cm).
- Enter three semi axes, any order.
- Halve full lengths first.
- Read volume first.
- Read equal sphere beside it.
- Multiply by density for mass.
Formula
- a,b,c
- Semi axes, half of full lengths.
- 4/3 pi
- Sphere constant, 4.18879.
- d
- Equal sphere diameter from volume.
Equal axes collapse to the sphere 4/3 pi r cubed automatically. Full diameters entered as semi axes inflate volume exactly 8 times, the classic error this page warns against.
Worked example
Example: semi axes 3, 4, 5
Defaults: imperial, semi axes 3, 4, 5 in.
- Product: 3 x 4 x 5 = 60.
- Volume: 4.18879 x 60 = 251.33 cu in.
- Sphere: cube root of 60 is 3.91, double 7.83.
- Tool shows 251.33 cu in, 7.83 in equal sphere.
- Sphere check 5,5,5: 523.60 cu in, 10 in.
- Metric mirror gives 4,118.52 cu cm, 19.89 cm.
Reference table
| Axes | Volume | Equal sphere |
|---|---|---|
| 3, 4, 5 | 251.33 cu in | 7.83 in |
| 5, 5, 5 | 523.60 cu in | 10 in |
| 2, 3, 4 | 100.53 cu in | 5.77 in |
Limits
- Ideal smooth shapes only.
- Hollow shells need wall math.
- Zero or negative axes rejected.
- Lumpy produce approximates.
Common mistakes
- Entering full lengths as semi axes.
- Mixing inches and centimeters.
- Ordering axes as if it mattered.
- Reading sphere where volume belongs.
- Skipping the 8 times sanity check.
Try it yourself
Quick checks. Answers sit tucked under each question.
Sphere 5, 5, 5?
523.60 cu in with a 10 in sphere.
Axes 2, 3, 4?
100.53 cu in with a 5.77 in sphere.
Diameter warning?
Full lengths inflate volume 8 times; halve first.
Questions
What is ellipsoid volume formula?
4/3 pi times the three semi axes multiplied.
Semi axis meaning?
Half of the full length along that direction.
Spheroid case?
Two axes equal still runs the same line.
Equal sphere use?
Compares odd shapes to familiar ball sizes.
Watermelon estimate?
Tape three half lengths, run this page, times density.
Planet volumes?
Same formula with equatorial and polar semi axes.