Quick answer: V = (4/3) π a² c, where a is the equatorial semi-axis and c the polar one. The same formula covers prolate and oblate — only the surface area differs. A rugby-ball shape 10 cm long and 6 cm across (a = 3, c = 5) holds 188.5 cm³. Set a = c and it collapses to the sphere, (4/3)πr³.
Prolate, oblate, and why one formula does both
A spheroid is an ellipse spun about one of its axes — an ellipsoid of revolution. Because it is a solid of revolution, every slice perpendicular to the spin axis is a perfect circle. That is what separates a spheroid from a general ellipsoid, where all three axes differ and the slices are ellipses.
| Shape | Semi-axes | Looks like | Volume |
|---|---|---|---|
| Prolate spheroid | a = b < c | Rugby ball, American football, cigar | (4/3)πa²c |
| Oblate spheroid | a = b > c | Discus, lentil, the Earth, M&M | (4/3)πa²c |
| Sphere | a = b = c | Ball | (4/3)πr³ |
| General ellipsoid | a ≠ b ≠ c | Squashed egg | (4/3)πabc |
The volume formula is the same for prolate and oblate because it never asks which axis is longer. Start from the general ellipsoid, V = (4/3)πabc, and set b = a: you get (4/3)πa²c whether c is bigger or smaller than a. Multiplication does not care about the order.
Worked example: a rugby ball 10 cm long, 6 cm across. Those are diameters, so halve them: c = 5, a = 3.
- a² = 9
- (4/3)π = 4.18879
- V = 4.18879 × 9 × 5 = 188.50 cm³
Sanity check: the enclosing box is 6 × 6 × 10 = 360 cm³, and 188.5 ÷ 360 = 0.5236 — which is π/6, exactly the fraction any ellipsoid fills of its bounding box. That ratio is constant regardless of how stretched or flattened the spheroid is, and it is a useful check on any answer.
Surface area is where they part company
Volume ignores which axis is longer. Surface area does not, because it depends on the curvature of the profile ellipse — and stretching curves differently from flattening.
S = 2πa² · (1 + (c ÷ (a·e)) · arcsin e)
Oblate (a > c), e = √(1 − c²/a²):
S = 2πa² + (πc² ÷ e) · ln((1 + e) ÷ (1 − e))
Both collapse to 4πr² when the eccentricity goes to zero, which is the check that they are right.
The awkwardness here is real and worth stating: there is no single elementary formula for spheroid surface area covering both cases, and for a general ellipsoid with three unequal axes there is no closed form at all — only approximations such as Knud Thomsen's. Volume is easy; surface area is genuinely hard. Anyone offering you one tidy surface-area formula for all ellipsoids is giving you an approximation, whether or not they say so.
The Earth is an oblate spheroid
The best-known spheroid of all, and a good demonstration of why the distinction matters.
| Quantity | Value |
|---|---|
| Equatorial radius a | 6,378.137 km |
| Polar radius c | 6,356.752 km |
| Difference | 21.385 km |
| Flattening f = (a − c)/a | 0.003353 (1 : 298.26) |
| Eccentricity e | 0.08182 |
| Volume as an oblate spheroid | 1.0832 × 10¹² km³ |
| Volume if treated as a sphere of radius a | 1.0868 × 10¹² km³ |
| Error from ignoring flattening | +0.34% (≈ 3.6 × 10⁹ km³) |
Twenty-one kilometres of difference across a body eight thousand miles wide is why the Earth looks perfectly round from orbit. But that third of a percent is enormous in absolute terms — about 3.6 billion cubic kilometres, some 4,500 times the volume of the Mediterranean Sea. It is also why every GPS fix, map projection and survey reference is computed on a reference ellipsoid such as WGS 84 rather than on a sphere: at the scale of the planet, an error you cannot see is still an error of kilometres on the ground.
The cause is rotation. Spin a fluid body and the equator is flung outward while the poles are drawn in. Faster rotation means more flattening — Jupiter, which turns in under ten hours, has a flattening of about 1 in 15 and is visibly squashed in any telescope.
Spheroids in the world
| Object | Type | a (equatorial) | c (polar) | Volume |
|---|---|---|---|---|
| Rugby ball | Prolate | 9.5 cm | 14.0 cm | 5,290 cm³ |
| American football | Prolate | 8.5 cm | 14.0 cm | 4,235 cm³ |
| Watermelon (oval) | Prolate | 12 cm | 18 cm | 10,857 cm³ |
| Hen's egg (approx.) | Prolate* | 2.1 cm | 2.9 cm | 53.6 cm³ |
| Lentil | Oblate | 0.3 cm | 0.1 cm | 0.0377 cm³ |
| M&M sweet | Oblate | 0.68 cm | 0.35 cm | 0.678 cm³ |
| Discus (men's) | Oblate | 11 cm | 2.2 cm | 1,115 cm³ |
| Earth | Oblate | 6,378 km | 6,357 km | 1.083 × 10¹² km³ |
| Jupiter | Oblate | 71,492 km | 66,854 km | 1.431 × 10¹⁵ km³ |
* An egg is not truly a spheroid — it is an ovoid, wider at one end than the other, so the spheroid model overestimates slightly. Dimensions here are representative rather than standardised.
Where this actually gets used day to day is medical imaging. The prolate spheroid is the standard model for the prostate, and the same π/6 coefficient drives kidney, spleen, thyroid and ovarian volume estimates. Three ultrasound diameters, one coefficient, and the organ is modelled as a spheroid — which is why π/6 = 0.5236 turns up on so many radiology reports.
Common mistakes & pro tips
- Entering diameters as semi-axes. The formula wants radii. Using diameters overstates the volume eightfold. The calculator has a diameter mode for exactly this reason.
- Assuming the surface area formula is symmetric. Volume is; surface area is not. Using the prolate expression on an oblate shape gives a wrong answer, and the functions involved may return NaN rather than failing loudly.
- Mixing up a and c. c is along the axis of rotational symmetry; a is perpendicular to it. Swapping them changes the volume unless the shape is a sphere.
- Calling a general ellipsoid a spheroid. If all three axes differ it is not a spheroid, and the a²c shortcut does not apply — use (4/3)πabc.
- Treating an egg as a spheroid. Eggs are ovoid, asymmetric end to end. The model runs a few percent high.
- Pro tip: the π/6 check. Every spheroid fills exactly 0.5236 of its bounding box, however stretched. If your answer divided by (2a × 2a × 2c) is not 0.5236, something is wrong.
- Pro tip: sanity-check against the sphere. Set a = c in your working. If it does not return (4/3)πr³, the rearrangement is wrong.
How to use this calculator
- Leave it on "detect automatically" and it will identify prolate or oblate from your inputs and apply the right surface-area formula.
- Enter the equatorial semi-axis a and polar semi-axis c, or switch to diameter mode if that is what you measured.
- Read the flattening and eccentricity to see how far the shape departs from a sphere.
- Check the bounding-box ratio — it should always be 0.5236.
Frequently asked questions
What is the formula for the volume of a spheroid?
Volume = (4/3)πa²c, where a is the equatorial semi-axis and c is the polar semi-axis. The same formula applies to both prolate and oblate spheroids. A prolate spheroid 10 cm long and 6 cm across has a = 3 and c = 5, giving a volume of 4.18879 × 9 × 5, which is 188.5 cm³.
What is the difference between a prolate and an oblate spheroid?
A prolate spheroid is stretched along its axis of symmetry, so the polar semi-axis is longer than the equatorial one. A rugby ball or an American football is prolate. An oblate spheroid is flattened along that axis, so the polar semi-axis is shorter. The Earth, a discus and a lentil are oblate. The volume formula is identical for both; only the surface area formula differs.
Is a spheroid the same as an ellipsoid?
A spheroid is a special case of an ellipsoid in which two of the three semi-axes are equal, so every cross-section perpendicular to the axis of symmetry is a circle. A general ellipsoid has three different semi-axes and elliptical cross-sections, with volume (4/3)πabc. Every spheroid is an ellipsoid, but most ellipsoids are not spheroids.
What is the volume of the Earth?
Treating the Earth as an oblate spheroid with an equatorial radius of 6,378.137 km and a polar radius of 6,356.752 km gives a volume of about 1.083 × 10¹² km³, or 1.083 trillion. Using the equatorial radius as if the Earth were a perfect sphere overstates that by about a third of one percent, which is roughly 3.6 billion cubic kilometres.
What is flattening?
Flattening measures how far a spheroid departs from a sphere. It is the equatorial radius minus the polar radius, divided by the equatorial radius. A sphere has a flattening of zero. The Earth's flattening is about 0.00335, usually written as 1 : 298.257, which is small enough that the Earth photographs as a circle but large enough to matter for satellite navigation and surveying.
Why is the surface area of a spheroid harder than the volume?
The volume comes from a straightforward integral that produces the clean (4/3)πa²c. The surface area requires integrating arc length around an ellipse, which brings in inverse trigonometric or inverse hyperbolic functions and gives different expressions for the prolate and oblate cases. There is no single elementary formula covering both, and for a general ellipsoid with three unequal axes there is no closed form at all.
References & further reading
- WGS 84 reference ellipsoid parameters — equatorial radius 6,378,137 m and flattening 1 : 298.257223563.
- Standard closed-form surface-area expressions for prolate and oblate spheroids.
- NASA planetary fact sheets — equatorial and polar radii for Earth and Jupiter.
- Thomsen's approximation for the surface area of a general triaxial ellipsoid.
Volume and surface area here are exact closed-form results for a true spheroid. Real objects are only approximately spheroidal — eggs, fruit and organs all depart from the model. See our accuracy policy.