Torus Volume Calculator (Volume & Surface Area)
By Santaji GadepatilinUpdated November 6, 2026
A torus is the doughnut shape made by sweeping a circle of radius r around an axis at distance R from its centre. This torus volume calculator gives the volume and surface area with Pappus's theorem, along with the inner and outer diameters — handy for O-rings, inner tubes and ring-shaped tanks.
How to use the torus volume calculator
Enter the major radius R — from the centre of the hole to the centre of the tube — and the minor radius r of the tube itself. The calculator returns the volume, the surface area and the inner and outer diameters of the ring. If you have diameters, R = (Douter + Dinner)/4 and r = (Douter − Dinner)/4.
Torus formulas
V = 2π² R r² · A = 4π² R r
These follow from Pappus’s centroid theorems: the volume is the cross-section area times the distance its centroid travels, and the area is the cross-section perimeter times that distance.
Worked example
R = 10 cm, r = 3 cm:
- V = 2π² × 10 × 9 = 1,776.53 cm³ (1.777 L)
- A = 4π² × 10 × 3 = 1,184.35 cm²
- Inner diameter = 14 cm, outer diameter = 26 cm
Everyday tori (approximate)
| Object | R | r | Volume | Surface area |
|---|---|---|---|---|
| Bagel | 4.5 cm | 2 cm | 355.3 cm³ | 355.3 cm² |
| Doughnut | 4 cm | 2.5 cm | 493.5 cm³ | 394.8 cm² |
| O-ring 100 × 5 mm | 5.25 cm | 0.25 cm | 6.5 cm³ | 51.8 cm² |
| Swim ring | 25 cm | 10 cm | 49,348.0 cm³ | 9,869.6 cm² |
| Car tyre inner tube | 30 cm | 9 cm | 47,966.3 cm³ | 10,659.2 cm² |
Frequently asked questions
What is the volume of a torus?
V = 2π²Rr², where R is the distance from the centre of the torus to the centre of the tube and r is the tube radius. It equals the tube's cross-section area πr² times the path length 2πR.
What is the surface area of a torus?
A = 4π²Rr — the tube circumference 2πr times the path length 2πR.
How do I get R and r from inner and outer diameters?
r = (outer − inner) ÷ 4 and R = (outer + inner) ÷ 4. An O-ring of 90 mm inner diameter and 5 mm section has r = 2.5 mm and R = 47.5 mm.
What if r is larger than R?
The tube overlaps itself (a spindle torus) and the formulas overcount the volume. They are valid for r ≤ R.