A torus is generated by revolving a circle around an axis that does not touch it, and its volume follows from Pappus centroid theorem: the volume of a solid of revolution equals the area of the generating shape times the distance its centroid travels. Here that gives two pi squared R r squared, which is the area of the small circle times the circumference of the path it sweeps.
The same theorem gives the surface area as the perimeter of the generating circle times the same travelled distance, four pi squared R r. Both results are exact, which is unusual for a curved solid, and they make the torus one of the easier three-dimensional shapes to work with despite its awkward appearance. The condition is only that the major radius exceeds the minor, so the hole exists.
The volume by major and minor radius table below carries 5 pre-computed rows, from 5 to 20, so a figure can be checked against this page without typing anything into it. It loads pre-filled, with major radius at 10, and the answer updates on every keystroke after that. The 4 bands (slim torus at one end, very large torus at the other) exist to answer the question a number on its own cannot: is this high, low or ordinary? Most visitors arrive here from, or leave for, Capsule Volume Calculator and Rectangular Prism Volume Calculator.
The formula
R- Major radius, centre of the hole to centre of the tube
r- Minor radius, the tube thickness
2πR- Distance travelled by the centroid of the tube cross-section
How it works, step by step
- Enter the major radius — from the centre of the hole to the centre of the tube.
- Enter the minor radius, which is half the tube thickness.
- The gauge shows the volume.
- The surface area, outer diameter and hole diameter appear below.
Worked examples
Major radius 10, minor radius 3
Volume is 2 pi squared × 10 × 9 = 1776.5288 and surface area is 4 pi squared × 10 × 3 = 1184.3525. The outer diameter is 26 and the hole is 14 across.
A car tyre approximation
A major radius of 30 cm and a minor of 10 cm gives 59,218 cubic cm, or about 59 litres of enclosed volume.
Checking the 10 row
The reference table lists Major R = 10, giving r = 1: 197.39, r = 2: 789.57, r = 3: 1,776.53, r = 4: 3,158.27. Entering that above should reproduce the same figures — if it does not, the input is being read differently to the way the table assumes.
How to read your score
Frequently asked questions
What is Pappus centroid theorem?
It states that the volume of a solid of revolution equals the generating area multiplied by the distance travelled by its centroid. For a torus that centroid travels 2 pi R.
What happens when the minor radius exceeds the major?
The hole closes and the shape self-intersects, becoming a spindle torus. The formulae above no longer describe the enclosed volume correctly.
How do I measure the major radius on a real object?
Take the outer and inner diameters, add them and divide by four. That gives the distance from the centre to the middle of the tube.
Is the surface area really exact?
Yes. Unlike the ellipse perimeter, the torus surface area has a clean closed form, because both generating curves are circles.
What do the bands under the volume mean?
They label the figure so it can be read without a reference point of your own: Slim torus from 0 to 300; Moderate torus from 300 to 1,200; Fat torus from 1,200 to 4,000; Very large torus from 4,000 upwards. The band is a reading aid only — the number itself is exact, and a value sitting just over a boundary is not meaningfully different from one just under it.
What formula does the Torus Volume Calculator use?
It computes V = 2π^2Rr^2; A = 4π^2Rr, where R is major radius, centre of the hole to centre of the tube; r is minor radius, the tube thickness; 2πR is distance travelled by the centroid of the tube cross-section. The same expression is printed on the page above the inputs, so the result can be reproduced by hand or in a spreadsheet.
Torus volume and area
| Major R | r = 1 | r = 2 | r = 3 | r = 4 | r = 5 |
|---|---|---|---|---|---|
| 5 | 98.70 | 394.78 | 888.26 | 1,579.14 | 2,467.40 |
| 8 | 157.91 | 631.65 | 1,421.22 | 2,526.62 | 3,947.84 |
| 10 | 197.39 | 789.57 | 1,776.53 | 3,158.27 | 4,934.80 |
| 15 | 296.09 | 1,184.35 | 2,664.79 | 4,737.41 | 7,402.20 |
| 20 | 394.78 | 1,579.14 | 3,553.06 | 6,316.55 | 9,869.60 |
Volume rises linearly with R but with the square of r.