The Inputs Are Half-Measurements
This calculator asks for three semi-axes: the distance from the center to the surface in three perpendicular directions. If you measured the object end to end, halve each measurement first.
An egg-like shape 20 cm long, 12 cm wide, and 8 cm high has a = 10, b = 6, c = 4. Entering the full dimensions instead makes the volume eight times too large, since each of the three factors is doubled. All three values must be in the same unit. If you only need to keep two of those dimensions in proportion while resizing, the Aspect Ratio Calculator does that directly.
V = (4/3)πabc
An ellipsoid is a sphere stretched or squashed along three directions. The sphere formula (4/3)πr³ multiplies r by itself three times; the ellipsoid formula swaps those three r's for three different semi-axes. Set a = b = c and you are back to the sphere volume formula.
The calculator also shows an equivalent radius, the cube root of abc. That is the radius of a sphere with exactly the same volume, which is a quick way to picture how big the shape really is.
The Ellipsoid Family
| Type | Semi-axes | Looks like | Example volume |
|---|---|---|---|
| Sphere | a = b = c | a ball | r = 5 cm: about 523.6 cm³ |
| Prolate spheroid | a = b < c | a rugby ball, a watermelon | 4, 4, 9 cm: 192π ≈ 603.2 cm³ |
| Oblate spheroid | a = b > c | the Earth, a lens | 8, 8, 4 cm: 341.33π ≈ 1,072.3 cm³ |
| Triaxial ellipsoid | all different | a pebble, an egg-like organ | 6, 4, 3 cm: 96π ≈ 301.6 cm³ |
One formula covers every row, so you don't need to decide which type you have. Just enter the three semi-axes. A shape with a straight cylindrical middle and rounded ends is not a spheroid, though. That's a capsule, handled by the Capsule Volume Calculator.
Worked Examples
From full dimensions: a 16 × 10 × 8 cm model
Halve to get a = 8, b = 5, c = 4. The product is 160, and (4/3)π × 160 ≈ 670.2 cm³.
From a ratio: semi-axes 2:1:1 with the short ones at 4 cm
A ratio alone doesn't fix the size, so you need at least one real measurement. Here a = 8, b = 4, c = 4, the product is 128, and V ≈ 536.2 cm³.
Which is bigger: a sphere of radius 5 cm or an ellipsoid of 7, 5, 3 cm?
The ellipsoid's longest semi-axis is bigger than the sphere's radius, but compare the products: 7 × 5 × 3 = 105 against 5 × 5 × 5 = 125. The sphere wins, 523.6 cm³ to 439.8 cm³. Stretching one direction doesn't make up for squashing another.
Ellipsoid Volume in Medical Imaging
Ultrasound, CT, and MRI reports often estimate the volume of an organ, cyst, or lymph node by treating it as an ellipsoid. The scan gives three perpendicular diameters, usually in millimeters. Take a structure measuring 30 × 18 × 12 mm:
- Semi-axes: 15, 9, and 6 mm; product 810
- V = (4/3)π × 810 ≈ 3,392.9 mm³, or about 3.39 mL
Clinicians often use a shortcut instead: multiply the three full diameters by 0.52. That is the same formula rearranged, since (4/3)π ÷ 8 = π/6 ≈ 0.5236. Rounding to 0.52 gives 30 × 18 × 12 × 0.52 ≈ 3,370 mm³, slightly lower than the exact figure. Real tissue is never a perfect ellipsoid, so treat either number as an estimate. Interpreting it is a job for a qualified professional.
Ellipse or Ellipsoid?
An ellipse is a flat oval with area but no volume: area = πab. An ellipsoid has depth in all three directions. If someone asks for "ellipse volume," they almost always mean the 3D shape.
Units
Semi-axes in millimeters give mm³, in meters give m³. Useful anchors: 1,000 mm³ = 1 mL, 1,000 cm³ = 1 liter, and 1 m³ = 1,000 liters. The calculator also converts to US and imperial gallons and cubic feet.
How the Numbers Are Checked
Volume uses the exact formula and carries no approximation. Surface area is different: for three unequal axes it has no closed-form solution, so the code hardcodes p = 1.6075 and runs the Knud Thomsen approximation, which is off by at most about 1.06% in the worst case. Both figures are shown to the same number of decimal places, but only the volume is exact.
Frequently Asked Questions
How do you calculate the volume of an ellipsoid?
Use V = (4/3)πabc, where a, b, and c are the three semi-axes (center-to-surface distances in three perpendicular directions). Multiply all three semi-axes together, multiply by π, then multiply by 4/3.
If all three semi-axes are equal, does this become the sphere formula?
Yes. When a = b = c = r, the formula V = (4/3)πabc becomes V = (4/3)πr³, the standard sphere volume formula. A sphere is just an ellipsoid with three equal semi-axes.
Why do radiologists use ellipsoid volume?
MRI and ultrasound scans give three perpendicular measurements of a mass or organ. Plugging those into the ellipsoid formula gives a reproducible volume estimate without complex 3D integration. It is an approximation, real biological shapes are not perfect ellipsoids, but it is widely accepted in clinical practice for structures like lymph nodes, tumors, and follicles.
References
- Ellipsoid, Wikipedia: geometric definition, volume and surface area formulas, and use in geodesy and medical imaging.
- Ellipsoid, Wolfram MathWorld: formal mathematical treatment of the ellipsoid including all three semi-axis cases.
- Ellipsoid, Math is Fun: approachable explanation of semi-axes and how the ellipsoid formula relates to the sphere formula.