A torus is exact; an ellipsoid is only half exact. Torus volume is 2π²Rr² and surface area 4π²Rr, both from Pappus. An ellipsoid’s volume is exactly (4/3)πabc, but its surface area has no elementary formula and can only be approximated.
Pappus and the doughnut
Both torus formulas come from one theorem, and it needs no calculus at all.
Pappus’ centroid theorem: spin a shape around an axis it does not cross, and the solid’s volume is the shape’s area times the distance its centroid travels.
For a torus, the shape is a circle of area πr² and its centroid travels 2πR. Multiply: 2π²Rr². That is the whole derivation.
The surface version is the same idea with the perimeter: 2πr times 2πR gives 4π²Rr.
What is genuinely surprising is that bending changes nothing. A torus has exactly the volume of a straight cylinder of length 2πR — the inside of the ring is compressed exactly as much as the outside is stretched, and the two cancel precisely.
The ellipsoid’s easy half
V = (4/3)πabc. Exact, and a clean generalisation of the sphere formula with the three semi-axes in place of r³.
It works for the same reason the ellipse area does: an ellipsoid is a sphere stretched by independent factors along three perpendicular directions, and stretching multiplies volume by the product of those factors.
Setting all three equal recovers (4/3)πr³ exactly, which is a good check on any implementation and one this page’s suite makes.
Two special cases have names. An oblate spheroid is squashed at the poles — the Earth is one. A prolate spheroid is stretched along one axis — a rugby ball.
The approximation
The surface area is a different matter entirely, and it is worth being blunt about it.
A general ellipsoid’s surface area has no elementary closed form. It needs elliptic integrals, exactly as an ellipse’s perimeter does — the same obstruction, one dimension up.
So every formula quoted for it is an approximation. The usual one is Knud Thomsen’s, and this page uses it.
Its worst-case relative error is about 1.06%. That is far larger than people expect from a formula printed without qualification, and it is why this page states it rather than presenting the number as a value.
Worth knowing about the formula itself: the exponent 1.6075 is not derived from anything. It was chosen empirically to minimise the worst-case error. That is a reasonable thing to do and a useful thing to know before relying on the result.
The special cases are exact. A sphere, and a spheroid with two equal axes, both have closed forms — it is only the general three-different-axes case that does not.
When the hole closes
A torus stops being a ring when the tube gets too fat, and the formulas stop describing it.
While r < R it is a ring torus — a doughnut with a hole. At r = R the hole closes to a point (a horn torus), and beyond that the tube passes through itself (a spindle torus).
The Pappus formulas assume the generating circle does not cross the axis, which is exactly the r < R condition. Past it, the volume formula double-counts the overlapping region and returns something that is not the volume of anything.
This page refuses r ≥ R with that reason, rather than returning a number that looks fine.
Where they get used
Seals and gaskets. An O-ring is a torus, and its volume determines the material and its compression behaviour.
Physics. Tokamak fusion reactors are toroidal, because a ring has no ends for the plasma to escape from.
Geodesy. The Earth is an oblate spheroid, flattened by about a part in 300. Every map projection and GPS calculation is built on that ellipsoid rather than on a sphere.
Biology and food. Eggs, seeds, cells and fruit are approximately ellipsoidal, and volume estimates from three measured axes are standard practice — with the surface area, as ever, being the harder half.
Sources and methodology
Pappus’ theorem and the ellipsoid approximation are standard; these are the references.
Method. The torus figures come from Pappus’ centroid theorem and are exact; the suite verifies them against direct numerical integration of annular cross-sections. The ellipsoid volume is exact and its surface area is not: it uses Knud Thomsen’s approximation, which is labelled as such and carries a stated worst-case error of about 1.06%. That claim is verified rather than repeated — the suite integrates the actual surface and confirms an oblate case lands at 1.01%, right against the bound, and a sphere at zero. That engine is verified on every change against 96 hand-written assertions, including that torus volumes match direct integration, and that Thomsen’s approximation stays inside its stated 1.06% bound on every ellipsoid tested — checked against numerical surface integration rather than taken on trust. The count and the per-case breakdown are published on the formula verification page.
Read the guide
The two-dimensional version of the same missing formula — an ellipse’s perimeter — is on the Ellipse Calculator.