Volume of a torus

The formula

• V = 2π²Rr²

R is the major radius, from the centre of the hole to the centre of the tube, and r the minor radius, the radius of the tube itself. A ring doughnut, a tyre and a lifebuoy are tori.

A worked example

R = 5 and r = 2 cm: V = 2π² · 5 · 4 = 40π² ≈ 394.8 cm³. A bicycle inner tube with R = 32 cm and r = 1.5 cm holds 2π² · 32 · 2.25 ≈ 1,420 cm³, about 1.4 litres of air.

Why a product

Cut the tube anywhere and the cross-section is a circle of area πr². Its centre travels round a circle of radius R, a distance of 2πR, and the volume is the area times that distance: πr² · 2πR = 2π²Rr². That is Pappus’s theorem, and it is the same reasoning as a cylinder, whose circle travels in a straight line instead.

Holds when

The formula needs R ≥ r, so that the tube does not pass through itself. At R = r the hole closes to a point, a horn torus, and the formula still holds; below it the solid overlaps itself and the product overcounts.

Common mistakes

• Measuring R to the outer edge. R runs to the middle of the tube. From the outer radius R₁ and the inner R₂, R = (R₁ + R₂) / 2 and r = (R₁ − R₂) / 2.
• Squaring the wrong radius. It is the tube radius r that is squared; R appears once.
• Using π² as 2π. π² ≈ 9.87, not 6.28.

Frequently asked questions