About the Torus Calculator
A torus is the donut shape you get by sweeping a circle around an axis that lies in the same plane but outside the circle. Doughnuts and bagels, inner tubes, O-rings, swim rings, lifebuoys, toroidal transformer cores and curved tube bends are all tori.
This torus calculator works from either the major radius R (centre of the hole to the centre of the tube) and the minor radius r (radius of the tube), or from the inner and outer radii you can measure directly with a ruler. It returns the volume, the surface area, both sets of radii and the tube diameter, plus the capacity in litres and US gallons when you choose a real length unit — handy for inflatable rings or O-ring material.
The formulas assume a ring torus, where the tube does not overlap itself (r < R). If r ≥ R the hole closes up and the shape self-intersects; the calculator flags that case.
With the default inputs, the volume is 1,776.5288. Change any value above to recalculate instantly.
How to use the torus calculator
- 1Choose whether you know R and r or the inner and outer radii.
- 2Enter the two radii (halve diameters first).
- 3Pick a length unit to get litres and gallons.
- 4Read the volume and surface area.
Formula and method
By Pappus’s centroid theorems, the volume of a solid of revolution equals the area of the rotated shape times the distance its centroid travels, and the surface area equals the perimeter times that distance. The tube’s circular cross-section has area πr² and perimeter 2πr, and its centre travels a full circle of length 2πR.
That gives V = πr² × 2πR = 2π²Rr² and A = 2πr × 2πR = 4π²Rr. When you enter inner and outer radii, the calculator converts them with R = (outer + inner)/2 and r = (outer − inner)/2. The formulas are exact for a ring torus (r < R); for a self-intersecting spindle torus they double-count the overlap.
- R
- Major radius: centre of the hole to the centre of the tube
- r
- Minor radius: radius of the tube
- V
- Volume
- A
- Surface area
Worked examples
R = 10, r = 3
V = 2 × π² × 10 × 3² = 180π² ≈ 1,776.53 cubic units and A = 4 × π² × 10 × 3 = 120π² ≈ 1,184.35 square units. The hole has a radius of 7 and the ring reaches out to 13.
Swim ring: 45 cm across outside, 20 cm hole
An inner radius of 10 cm and outer radius of 22.5 cm give R = 16.25 cm and r = 6.25 cm. The ring holds 2π² × 16.25 × 6.25² ≈ 12,530 cm³ — about 12.5 litres of air — and has about 4,010 cm² of vinyl surface.
Doughnut: 4.5 cm outer radius, 1.5 cm hole radius
R = (4.5 + 1.5)/2 = 3 cm and r = (4.5 − 1.5)/2 = 1.5 cm, so the doughnut’s volume is 2π² × 3 × 2.25 ≈ 133.2 cm³ and its glazeable surface is 4π² × 3 × 1.5 ≈ 177.7 cm².
Frequently asked questions
What is the volume of a torus?+
V = 2π²Rr², where R is the distance from the centre of the hole to the centre of the tube and r is the tube radius. It is the same as a cylinder of radius r bent into a ring of length 2πR.
What is the surface area of a torus?+
A = 4π²Rr. For R = 10 and r = 3 that is 120π² ≈ 1,184.4 square units. It is the tube’s circumference 2πr multiplied by the distance 2πR travelled by its centre.
How do I measure R and r on a real ring?+
Measure the outer diameter and the hole diameter and halve them to get outer and inner radii. Then R = (outer + inner)/2 and r = (outer − inner)/2, or simply choose the inner/outer option in this calculator.
What happens when r is bigger than R?+
If r = R the hole shrinks to a single point (a horn torus); if r > R the tube passes through itself (a spindle torus). The standard formulas then count the overlapping part twice, so they overstate the enclosed volume.
How do I calculate O-ring volume?+
An O-ring is a torus: use the cross-section diameter as 2r and the mean diameter (inner diameter + cross-section) as 2R. A 20 mm ID, 3 mm cross-section ring has R = 11.5 mm and r = 1.5 mm, so V ≈ 510.8 mm³.