Hong Life CalculatorsUnderstanding the torus calculator — concept and principle
A torus is the doughnut shape you get by sweeping a circle around an axis that lies in the same plane but does not cross the circle. It is described by two numbers: the major radius R, from the centre of the hole to the centre of the tube, and the minor radius r, the radius of the tube itself. The volume and surface area are used for O-rings, inner tubes, ring tanks and toroidal cores.
How it is calculated
- Volume: V = 2π²Rr².
- Surface area: A = 4π²Rr.
Use one length unit for both inputs. The outer diameter of the torus is 2(R + r) and the hole diameter is 2(R − r).
The principle
Both formulas come from Pappus's centroid theorems. When a flat shape is revolved around an external axis, the volume swept out equals the shape's area times the distance travelled by its centroid. Here the shape is a circle of area πr², and its centre travels a circle of length 2πR, so V = πr² × 2πR = 2π²Rr². The second theorem works the same way for curves: the surface area equals the length of the curve times the distance travelled by its centroid, so A = 2πr × 2πR = 4π²Rr.
An intuitive way to see it: cut the torus and straighten it. It becomes roughly a cylinder of radius r and length 2πR. The inner side gets squeezed and the outer side stretched by exactly balancing amounts, so the straight-cylinder formulas give the exact answer.
Worked example
- Inputs: R = 5, r = 1.
- π² ≈ 9.8696044.
- Volume: 2 × 9.8696044 × 5 × 1² ≈ 98.69604401.
- Surface area: 4 × 9.8696044 × 5 × 1 ≈ 197.392088.
| Minor radius r | Volume |
|---|---|
| 0.25 | 6.168502751 |
| 0.5 | 24.674011 |
| 0.75 | 55.51652476 |
| 1 | 98.69604401 |
| 1.25 | 154.2125688 |
| 1.5 | 222.066099 |
| 2 | 394.784176 |
How to read the result
Volume is proportional to R and to r², so a thicker tube adds volume much faster than a wider ring. Doubling r quadruples the volume, while doubling R only doubles it. The surface area is proportional to both R and r. For the example, the outer diameter is 12 and the hole diameter is 8. If the result is in cm³, divide by 1,000 to get litres.
Common mistakes
- Entering diameters instead of radii, which makes the volume eight times too large.
- Measuring R to the outer edge rather than to the centre of the tube. If you know the outer and inner diameters D and d, then R = (D + d) ÷ 4 and r = (D − d) ÷ 4.
- Using r larger than R. That is a self-intersecting (spindle) torus, and the formulas then count the overlapping region twice.
Frequently asked questions
How do I find R and r from a measured ring?
Measure the outer diameter D and the hole diameter d. Then r = (D − d) ÷ 4 and R = (D + d) ÷ 4. For D = 12 and d = 8, r = 1 and R = 5.
Is a torus volume the same as a bent cylinder?
Yes, exactly. By Pappus's theorem, a cylinder of radius r and length 2πR has the same volume as the torus.
What is the volume of an O-ring?
Treat it as a torus: r is half the cord thickness and R is the mean ring radius, the inner radius plus r.
Sources: Pappus of Alexandria, Collection, Book VII (centroid theorems for solids and surfaces of revolution) · J. Stewart, Calculus: Early Transcendentals, Cengage, section on volumes of revolution and Pappus's theorem