What this calculator does
A torus is the ring shape traced by a circle swept around an axis outside itself: an inner tube, a doughnut or a ring all share this shape. It needs two radii to describe: the ring radius, from the centre of the whole shape to the centre of the tube, and the tube radius, the thickness of the tube itself.
Both volume and surface area follow the same pattern as sweeping any circle through a circular path: multiply the tube’s own area or circumference by the distance that circle’s centre travels, which is 2π times the ring radius.
The formula
This treats the torus as Pappus’s theorem does: the volume equals the tube’s cross-sectional area, πr², times the distance its centre travels sweeping around, 2πR: giving 2π²Rr². Surface area follows the same logic with the tube’s circumference, 2πr, in place of its area, giving 4π²Rr. The construction only works when R is larger than r; if the tube radius exceeds the ring radius, the tube overlaps itself at the centre and the shape is no longer a simple ring.
| Term | Meaning |
|---|---|
| R | Ring radius: centre of the whole shape to the centre of the tube. |
| r | Tube radius: the thickness of the tube itself. |
| V | Volume, equal to 2π²Rr². |
| Outer / inner diameter | The overall width across the outside, and the width of the central hole, equal to 2(R + r) and 2(R − r). |
The inputs explained
| Field | What to enter |
|---|---|
| Ring radius (centre to tube centre) | The distance from the centre of the whole ring to the centre of the tube. |
| Tube radius | The radius of the tube itself. Must be smaller than R for a normal ring shape. |
When to use it
Volume of an inner tube, gasket or O-ring
Rubber seals, tyres and O-rings are torus-shaped. Volume here gives the material used; surface area gives what a coating or plating process needs to cover.
Sizing a doughnut-shaped tank or pipe loop
A circular pipe loop or ring-shaped tank uses the same geometry: the ring radius is the loop’s overall size, the tube radius is the pipe’s own radius.
Checking whether a ring shape is geometrically valid
If the tube radius is set larger than the ring radius, the inner diameter goes negative, which signals the tube would overlap itself: not a valid simple torus.
Estimating material for a decorative or architectural ring form
Circular handrails, ring-shaped light fittings and architectural mouldings that curve back on themselves are all toroidal, and the same volume and surface figures apply.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How volume changes as the ring widens, tube radius fixed at 3
A fixed tube thickness swept around rings of increasing size.
| Ring radius | Volume | Surface area | Outer diameter |
|---|---|---|---|
| 4 | 710.612 | 473.741 | 14.000 |
| 6 | 1,065.92 | 710.612 | 18.000 |
| 10 | 1,776.53 | 1,184.35 | 26.000 |
| 15 | 2,664.79 | 1,776.53 | 36.000 |
| 20 | 3,553.06 | 2,368.71 | 46.000 |
How volume changes as the tube itself thickens, ring radius fixed at 10
A fixed overall ring size with the tube growing thicker.
| Tube radius | Volume | Surface area | Inner diameter |
|---|---|---|---|
| 1 | 197.392 | 394.784 | 18.000 |
| 2 | 789.568 | 789.568 | 16.000 |
| 3 | 1,776.53 | 1,184.35 | 14.000 |
| 5 | 4,934.80 | 1,973.92 | 10.000 |
| 8 | 12,633.09 | 3,158.27 | 4.000 |
Questions
What is the difference between the ring radius and the tube radius?
The ring radius (R) is the distance from the centre of the whole shape out to the centre of the tube: roughly the size of the doughnut. The tube radius (r) is the thickness of the tube itself, the size of a cross-section cut through it.
What happens if the tube radius is larger than the ring radius?
The tube would have to overlap itself through the centre, which is not a simple torus and produces a negative inner diameter: a sign the inputs describe an impossible ring shape.
How do I find the volume of a torus?
Multiply the tube’s cross-sectional area, πr², by the distance its centre sweeps around the ring, 2πR: the two together give 2π²Rr², this calculator’s volume formula.
What are the outer and inner diameters?
The outer diameter is the total width across the ring at its widest, R + r on each side. The inner diameter is the width of the central hole, R − r on each side: both doubled to give the full diameter.
Is this the same method used for a torus’s volume in general?
Yes: it applies Pappus’s centroid theorem, which finds the volume or surface of any shape swept in a circle by multiplying the swept shape’s own area or perimeter by the distance its centre travels.
For the circular cross-section this shape is built from, see the circle calculator. For a straight rather than looped tube, use the cylinder calculator.