What is the formula for the volume of a torus?

The volume of a torus is V = 2π²Rr², where R is the major radius (distance from the torus center to the tube center) and r is the minor radius (radius of the tube). This can also be written as V = (π × r²) × (2πR), i.e., the cross-sectional area times the circumference traced by the center of the tube.

What is the difference between major radius R and minor radius r?

The major radius R is the distance from the center of the entire torus to the center of the circular tube. The minor radius r is the radius of that circular tube itself. For a donut, R would be roughly the distance from the hole center to the middle of the dough, while r is half the thickness of the dough ring.

Does R have to be larger than r?

For a standard ring torus (with a hole in the middle), R must be greater than r. If R equals r, you get a horn torus (no hole, pinched at one point). If R is less than r, you get a spindle torus (self-intersecting). This calculator accepts all positive values but will warn if R < r.

What units does this calculator support?

You can enter dimensions in millimeters (mm), centimeters (cm), meters (m), inches (in), or feet (ft). The result is displayed in the matching cubic unit, plus conversions to other common units.

How accurate is the result?

The calculator uses JavaScript's built-in Math.PI constant (15+ significant digits) and rounds the final display to 6 significant figures, which is more than sufficient for engineering and academic use.

Volume of a Torus Calculator

Name: Volume of a Torus Calculator
Availability: InStock
Author: Nham Vu

Enter the major radius R and minor radius r to instantly calculate the volume of a torus using V = 2π²Rr².

Torus Dimensions

V = 2π²Rr²

Unit

Major radius R (center of torus to center of tube)

Minor radius r (radius of the tube)

Enter R and r, then click Calculate Volume.

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Summary

Enter the major radius R and minor radius r to instantly calculate the volume of a torus using V = 2π²Rr².

How it works

Enter the major radius R — the distance from the center of the torus to the center of the tube.
Enter the minor radius r — the radius of the tube (cross-section circle).
Select your preferred unit (mm, cm, m, in, ft).
The calculator instantly applies the formula V = 2π²Rr² to compute the volume.
Results are shown in the selected unit cubed and also converted to common units.

Use cases

Engineering design of O-rings, gaskets, and seals.
Architecture and industrial design involving toroidal structures.
Academic geometry and calculus homework.
Manufacturing estimation for ring-shaped components.
3D printing volume and material cost estimation.
Physics problems involving toroidal shapes.

Frequently Asked Questions

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Last updated: 2026-05-23 · Reviewed by Nham Vu