Last updated: 2026-05-18

Definition

A torus is the surface generated by revolving a circle around an axis in its plane that does not intersect the circle. The result is the familiar donut shape, also seen in O-rings, tire inner tubes, life preservers, and bagels. Two radii fully define it: the major radius R, the distance from the center of the torus to the center of the tube, and the minor radius r, the radius of the tube itself. The volume and surface area follow from these two radii alone.

Formulas

Both formulas follow directly from Pappus's centroid theorem: the volume (or surface area) swept by a shape revolving around an external axis equals the shape's area (or perimeter) multiplied by the distance its centroid travels.

Volume — the circular cross-section ($\pi r^2$) travels a distance of $2\pi R$:

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

Surface area — the circular perimeter ($2\pi r$) travels a distance of $2\pi R$:

$A = 4\pi^2 R r$

Types of torus

The relationship between R and r gives three distinct shapes:

This calculator returns the mathematically correct volume and surface area for all three cases. Only ring tori are physically achievable without self-intersection.

Worked example

Problem: A rubber O-ring has a major radius of 20 mm and a minor radius of 3 mm. Find its volume and surface area.

R = 20 mm = 0.020 m, r = 3 mm = 0.003 m:

$V = 2\pi^2 \times 0.020 \times 0.003^2 = 2 \times 9.8696 \times 0.020 \times 0.000009 \approx 3.553 \text{ cm}^3$

$A = 4\pi^2 \times 0.020 \times 0.003 = 4 \times 9.8696 \times 0.000060 \approx 23.69 \text{ cm}^2$