This ellipsoid volume calculator returns the volume of any ellipsoid from its three semi-axes using V = ⅔πabc, or four-thirds pi times a times b times c. That formula is exact. Semi-axes of 3, 2 and 1 cm give V = ⅔π × 6 = 8π = 25.13 cm³, which is 0.0251 litres, and the working line below the results shows that substitution with your own figures in it.
Surface area is a different story, and Arb Digital would rather say so on the page than quietly hide it. An ellipsoid with three different axes has no exact surface area formula in elementary functions at all: the true value is an elliptic integral. The figure this tool reports uses Knud Thomsen's approximation, which is accurate to within about 1.06% in the worst case. Where two axes are equal the tool switches to the exact spheroid formula instead and labels the result as exact.
What This Ellipsoid Calculator Does
An ellipsoid is the three-dimensional analogue of an ellipse: a closed surface where every plane slice through it is an ellipse or a circle. It is described by three semi-axes, a, b and c, each measured from the centre out to the surface along a perpendicular direction. Rugby balls, watermelons, gemstone cabochons, pressure vessel end caps, mineral grains, cell nuclei and the Earth itself are all modelled as ellipsoids.
The calculator takes those three semi-axes, or the full widths if that is what you physically measured, and returns the exact volume, a surface area with its accuracy stated, the radius of the sphere holding the same volume, the sphericity, and the classification of the solid as a sphere, a prolate spheroid, an oblate spheroid or a triaxial ellipsoid. Volume also appears in litres and US gallons, which is what you want if the ellipsoid is a tank end or a container rather than a solid body.
Sphericity is worth a word because it is a genuinely useful number rather than decoration. It is the surface area of the equal-volume sphere divided by the actual surface area of the solid, a measure introduced by Hakon Wadell in sedimentology. A sphere scores exactly 1, and everything else scores below 1 because no shape encloses a given volume in less surface than a sphere does. The 3-2-1 example scores 0.847.
How to Use It
- Decide whether you have semi-axes or full widths. Callipers across a whole object give full axis lengths; a drawing usually gives semi-axes.
- Enter the three values. Order does not matter. The volume formula is symmetric in a, b and c, and the tool sorts them internally to classify the shape.
- Set the unit. Volume comes back cubed and area squared in that unit, with litres and gallons converted underneath.
- Read the surface area label. It tells you whether the figure shown is exact or an approximation, and which one.
- Check the working line against the numbers you meant to enter before using the result for anything.
The Formula: How an Ellipsoid Is Calculated
With semi-axes a, b and c:
- Volume: V = ⅔πabc. For a = 3, b = 2 and c = 1 cm the product abc is 6, so V = ⅔ × π × 6 = 8π = 25.13 cm³.
- Equal-volume sphere radius: r = ∛(abc). Here ∛6 = 1.817 cm.
- Surface area, Thomsen's approximation: S ≈ 4π((aᵢbᵢ + aᵢcᵢ + bᵢcᵢ) / 3)¹⁄ᵢ with p = 1.6075. For 3, 2 and 1 that gives 48.97 cm².
- Sphericity: Ψ = 4πr² / S. That is 41.49 / 48.97 = 0.847.
The volume formula is easy to remember by scaling. Start from a unit sphere of volume ⅔π, then stretch it by a in one direction, b in the second and c in the third. Volume scales by the product of the three stretch factors, so V = ⅔πabc. Set a = b = c = r and it collapses to the familiar ⅔πr³, which is the case the sphere volume calculator handles directly. Take any single cross-section through the centre and you get an ellipse of area πab, which is where the circle calculator stops being enough.
Why Ellipsoid Surface Area Has No Exact Formula
The volume of an ellipsoid falls out of a simple change of variables, which is why it is clean. Surface area does not. Working out the area of the stretched surface requires integrating a square root that, for three unequal axes, cannot be reduced to elementary functions. The exact answer is written in terms of incomplete elliptic integrals of the first and second kind, and those are defined by the integral itself rather than by any combination of powers, roots, logarithms or trigonometric functions.
This is the same obstruction that stops the perimeter of a plain two-dimensional ellipse having a neat closed form, one dimension up. It is not a gap in anyone's knowledge and it will not be solved by a cleverer rearrangement. Any tool that prints an ellipsoid surface area to eight decimal places with no caveat is either evaluating an elliptic integral numerically or, far more often, printing an approximation and not telling you. The full mathematical treatment is set out at Wolfram MathWorld's ellipsoid entry.
Knud Thomsen's Approximation and How Accurate It Is
The formula used here is S ≈ 4π((aᵢbᵢ + aᵢcᵢ + bᵢcᵢ)/3)¹⁄ᵢ with the exponent p fixed at 1.6075. It is a power mean of the three axis-pair products, and the exponent is a fitted constant chosen to minimise the worst-case error rather than a value derived from theory.
Its behaviour is worth knowing precisely. The relative error never exceeds about 1.061% for any ellipsoid, and it is far smaller than that for most. It is exactly correct when all three axes are equal, because then the formula reduces to 4πr². It stays within a fraction of a percent for the mildly stretched shapes that make up nearly all practical work, and it only approaches its error ceiling for extreme, needle-like or disc-like proportions.
What that means in use: treat the surface area figure as reliable to two or three significant figures, and never as an exact value. If you are ordering material to coat an ellipsoidal surface, a 1% band is usually well inside the wastage allowance you would add anyway. If you need better than 1%, you need a numerical evaluation of the elliptic integral, not a closed-form shortcut.
Spheres and Spheroids: The Cases That Are Exact
Two special cases escape the problem entirely, and the calculator detects both and switches to the exact formula automatically.
A sphere has a = b = c. Volume is ⅔πr³ and surface area is exactly 4πr², with no approximation anywhere. A spheroid, also called an ellipsoid of revolution, has exactly two equal axes, and it comes in two flavours. A prolate spheroid is stretched along one axis, so two axes are equal and the third is longer: a rugby ball, an American football, a grain of rice. Its exact surface area is 2πa²(1 + (c/ae)·arcsin e), where a is the shared shorter semi-axis, c is the long one and e = √(1 − a²/c²) is the eccentricity of the generating ellipse.
An oblate spheroid is squashed along one axis, so two axes are equal and the third is shorter: a lentil, a discus, the Earth. Its exact surface area is 2πa² + π(c²/e)·ln((1 + e)/(1 − e)), with e = √(1 − c²/a²). Both expressions are elementary because a surface of revolution reduces the double integral to a single one that happens to be solvable. Only the fully triaxial case, with three genuinely different axes, resists.
You can watch the approximation's quality directly here. Enter 2, 2 and 4 and the tool reports the exact prolate value of 85.914 cm²; Thomsen's estimate for the same solid is 85.863, low by 0.06%. Enter 4, 4 and 2 and the exact oblate value is 138.750 against an estimate of 138.947, high by 0.14%. Neither would change a material order.
Where Ellipsoid Volume Actually Gets Used
The most common real use is estimating the volume of an object that is roughly egg-shaped from three measured widths. Measure the longest dimension, the widest dimension perpendicular to it and the depth perpendicular to both, halve each to get semi-axes, and V = ⅔πabc gives a workable volume without displacement or immersion. The approximation is only as good as the object's resemblance to a true ellipsoid, so it works well on smooth rounded bodies and badly on anything with a flat face or a dent.
Pressure vessel and tank end caps are frequently hemi-ellipsoidal, which is half an ellipsoid, so their capacity is half of ⅔πabc. The cylindrical barrel between two such caps is a separate calculation that the cylinder volume calculator covers, and the two volumes simply add. Sedimentology and materials science use the sphericity figure to describe how close a grain is to spherical, which affects packing density and flow behaviour. Any capacity conversion beyond litres and gallons is handled by the volume converter, and the cube root behind the equal-volume radius by the cube root calculator. Unit definitions on this page follow NIST's guide to the SI.
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Browse All Free Tools Talk To Arb DigitalCommon Mistakes to Avoid
- Entering full widths as semi-axes. That doubles all three values and the volume comes out eight times too large. Use the selector rather than halving by hand.
- Treating the surface area as exact. For three unequal axes it is an approximation with a stated error band, and rounding it to six decimals does not make it truer.
- Applying a spheroid formula to a triaxial solid. The prolate and oblate expressions need exactly two equal axes. They are wrong when all three differ.
- Mixing units between axes. One axis in inches and two in centimetres produces a number that means nothing. Convert first.
- Modelling a dented or flat-faced object as an ellipsoid. The formula assumes a smooth ellipsoidal surface, and it will overstate the volume of anything visibly flattened.
Related Free Tools From Arb Digital
The sphere volume calculator handles the equal-axis case with exact figures throughout, and the circle calculator covers the two-dimensional cross-section. For straight-sided solids the cylinder volume calculator is the closer match, the volume converter moves capacity between units, and the scientific calculator is there if you want to reproduce the fractional powers in Thomsen's formula step by step.
Frequently Asked Questions
Volume is four thirds of pi multiplied by the three semi-axes, written V = 4/3 pi a b c. Semi-axes of 3, 2 and 1 centimetres give 8 pi, or 25.13 cubic centimetres. This formula is exact.
Not in elementary functions when all three axes differ. The exact value is an elliptic integral. This page reports Knud Thomsen's approximation instead and labels it as an approximation rather than presenting it as exact.
Its relative error never exceeds about 1.061 percent for any ellipsoid, and it is much smaller for ordinary proportions. It is exactly correct for a sphere. Treat the result as good to two or three significant figures.
Both have exactly two equal axes. A prolate spheroid is stretched along the third axis, like a rugby ball. An oblate spheroid is flattened along it, like a lentil or the Earth. Both have exact surface area formulas.
Either. A semi-axis runs from the centre to the surface, so it is half the full width. Use the selector at the top to switch, because entering full widths as semi-axes makes the volume eight times too large.
It is the surface area of the sphere with the same volume divided by the actual surface area. A sphere scores 1 and every other shape scores less, because no solid encloses a volume in less surface than a sphere.
Yes for volume. A hemi-ellipsoidal end cap holds half of the full ellipsoid volume, so calculate the whole solid and divide by two. The surface area figure would need the flat elliptical face handled separately.
Results are mathematical values rounded for display, and the surface area for a triaxial ellipsoid is an approximation with a stated error band. Real objects have surface texture and manufacturing tolerance, so allow a margin before ordering material to these figures.
