Case 1
- Major radius R
- 10
- Minor radius r
- 3
1,776.528792
Open with these valuesTorus Volume Calculator
1,776.528792units³
Result: 1,776.528792 units³2π²Rr², which is just the tube's cross-section πr² carried once round the circle of length 2πR. R runs from the centre of the hole to the middle of the tube; r is the tube itself. When r reaches R the hole closes and the formula stops applying beyond that.
Worked examples
Case 2
- Major radius R
- 5
- Minor radius r
- 1
98.696044
Open with these valuesCase 3
- Major radius R
- 20
- Minor radius r
- 5
9,869.604401
Open with these valuesHow it's calculated
V = 2 × π² × R × r²
- StepMeasure from the centre of the hole to the middle of the tube — that is R.
- StepMeasure the radius of the tube itself — that is r.
- ResultRead the volume in the cube of the unit you entered.
Reference table
| R, r | Exact | Volume |
|---|---|---|
| 5, 1 | 10π² | 98.696044 |
| 8, 2 | 64π² | 631.654682 |
| 4, 4 | 128π² | 1263.309363 |
| 10, 3 | 180π² | 1776.528792 |
| 20, 5 | 1000π² | 9869.604401 |
Questions
How do you calculate the volume of a torus?
2π² times the major radius times the minor radius squared. R of 10 and r of 3 give 180π², about 1776.53 cubic units.
Which radius is which?
The major radius R reaches from the centre of the hole to the middle of the tube; the minor radius r is the tube's own thickness. The outside edge sits at R + r.
Why π squared?
Two circles are involved: the tube's cross-section contributes one π, and the circular path its centre travels contributes the other. That is Pappus's theorem.
What if the hole closes up?
At r = R the hole shrinks to a point — the last row but one in the table. Beyond that the tube overlaps itself and the formula no longer describes a real solid.
Sources and last check
Information, not professional advice.