How to Find the Volume of a Torus

A torus is the doughnut shape swept out by a circle turning about an axis at a distance from it. Its volume is 2 × π² × center distance × tube radius², by the theorem of Pappus. The center distance runs from the center of the doughnut to the center of the tube and must exceed the tube radius.

Explanation

A torus is what a circle sweeps out when it turns about an axis lying outside it: a doughnut, a swim ring, an O-ring.

V = 2\pi^2 R r^2

$V$ — the volume
$R$ — the center distance, from the center of the doughnut to the center of the tube
$r$ — the radius of the tube

$R$ must be greater than $r$ . If it is not, the tube passes through itself and the formula no longer holds.

The theorem of Pappus

The volume is the area of the shape that was turned, multiplied by the distance its centroid travelled. The cross-section of the tube is a circle of area $\pi r^2$ , and its center travels once around a circle of radius $R$ , a distance of $2\pi R$ .

V = \pi r^2 \times 2\pi R = 2\pi^2 R r^2

Example

With the defaults, a center distance of $R = 10$ and a tube radius of $r = 3$ .

V = 2\pi^2 \times 10 \times 3^2 = 180\pi^2 \approx 1776.5288

Measuring it

$R$ reaches the center of the tube, not the edge of the hole. Measuring from the outside, with an outer diameter $D$ and a tube diameter $d$ , gives $R = (D - d) \div 2$ and $r = d \div 2$ . The hole itself has diameter $2(R - r)$ .