Torus Volume Calculator
A torus is a doughnut (ring) shape.
Enter Values
How to use this calculator
- Enter the major radius R — the distance from the centre of the torus to the centre of the tube.
- Enter the minor radius r — the radius of the tube itself (its cross-section).
- Press Calculate for the volume and surface area.
How it works
A torus is a doughnut (ring) shape. Its volume is 2 × π² × R × r², where R is the major radius (centre of the ring to centre of the tube) and r is the minor radius (the tube's own radius). This equals the tube's circular cross-section area (π × r²) swept around the ring's centre circle of circumference 2 × π × R — a result of Pappus's theorem.
The surface area follows the same idea: 4 × π² × R × r, which is the tube's circumference (2 × π × r) swept around the same centre circle. Both formulas assume a perfect ring torus where r is no larger than R, so the tube does not overlap itself at the centre.
Worked example
Major radius R = 3 m, minor radius r = 1 m. Volume = 2 × π² × 3 × 1² = 59.2176 m³, and surface area = 4 × π² × 3 × 1 = 118.4353 m².
Common mistakes
- Swapping the two radii. R is the big centre-to-tube distance; r is the small tube radius. Entering them the other way gives the wrong volume.
- Using a diameter instead of a radius. Both inputs are radii — halve any diameter before entering it.
- Entering r larger than R. A real ring torus needs r ≤ R, otherwise the tube would overlap itself through the middle.
Frequently asked questions
What are the major and minor radius?
The major radius R is the distance from the centre of the whole torus to the centre of the tube. The minor radius r is the radius of the tube's circular cross-section. Together they define the doughnut's size and thickness.
What is the formula for the volume of a torus?
Volume = 2 × π² × R × r², where R is the major radius and r is the minor radius. It comes from sweeping the tube's cross-section area (π × r²) around the ring's centre circle (circumference 2 × π × R).
Can the minor radius equal the major radius?
Yes. When r = R the hole in the middle just closes to a point (a horn torus). If r were larger than R the tube would overlap itself, which is not a standard ring torus, so this calculator asks for r ≤ R.
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