Torus Volume Calculator: V = 2 pi^2 R r^2

The volume of a ring shape such as a tyre, an O-ring or a doughnut.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Choose which quantity you want to find.

Enter the other values in the units shown.

The calculator gives the answer with the formula, the substituted numbers, and the answer in other units.

Formulas used

  • V = 2\pi^2Rr^2
  • r = \sqrt{\frac{V}{2\pi^2R}}

Worked example

With the default numbers in the calculator above:

  1. The formula, arranged to find the volume: volume = 2 x pi^2 x major radius (centre of the ring to the centre of the tube) x minor radius (radius of the tube)^2
  2. Put the values in (every value in its SI unit): volume = 2 x pi^2 x 0.5 x 0.1^2
  3. The answer: volume: 0.098696 m3
  4. The same answer in other units: 98.696 litres; 98696 cm3

Change the numbers in the calculator to see the full working, every step, for your own case.

Common questions

What is the formula for the volume of a torus?

2 pi squared times the major radius times the square of the minor radius.

What is the difference between the major and minor radius?

The major radius goes from the centre of the ring to the centre of the tube, and the minor radius is the radius of the tube itself.

How is it derived?

It is the area of the circular cross-section (pi r squared) times the distance travelled by its centre (2 pi R).

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any mathematics text (volume of a torus, Pappus's theorem).

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