Torus Volume Calculator
The volume of a torus — a doughnut, an O-ring, an inner tube — from the ring radius and the tube radius.
A torus is a circle swept around another circle, and its volume is 2π²Rr².
How the torus volume calculator works
A torus is a circle swept around another circle, and its volume is 2π²Rr². The neat way to see it is Pappus's theorem: the volume of a solid of revolution equals the area of the shape being swept (πr²) times the distance its centre travels (2πR). Multiply those and the formula falls out.
Formula: cone ⅓πr²h; pyramid ⅓lwh; torus 2π²Rr²; hemisphere ⅔πr³; ellipsoid ⁴⁄₃πabc
Worked examples
| Inputs | Volume | Note |
|---|---|---|
| Ring 10, tube 3 | 1,776.5288 | 1,776.5 |
| An O-ring, ring 12 mm, tube 2 mm | 947.482 | 947.5 mm³ |
| A doughnut, ring 4 cm, tube 2 cm | 315.8273 | 315.8 cm³ |
FAQFrequently asked questions
What is the volume of a torus?
2π²Rr², where R is the ring radius and r the tube radius.
What is the ring radius?
From the centre of the hole to the centre of the tube — not to the outer edge.
What is Pappus's theorem?
The volume of a solid of revolution is the swept area times the distance its centroid travels. For a torus that is πr² × 2πR.
How wide is the whole doughnut?
Twice the ring radius plus twice the tube radius, so 2(R + r) across.
Why must the tube be smaller than the ring?
Otherwise the hole closes and it is no longer a torus — the formula stops describing the shape.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Australian Curriculum (ACARA) — Mathematics — the terms and methods taught in Australian schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.