What is the formula for the volume of a torus?

For a standard ring torus with major radius R (distance from the center of the tube to the center of the torus) and minor radius r (radius of the tube), the volume is V = 2π²Rr².

How do I compute torus volume from inner and outer radius?

If you know the inner radius Rin and outer radius Rout of the torus, you can compute the major and minor radii as R = (Rout + Rin) / 2 and r = (Rout − Rin) / 2, then use V = 2π²Rr².

What units can I use with the torus volume calculator?

You can enter the radii in millimeters, centimeters, meters, inches, or feet. The calculator converts internally to cubic meters and reports results in m³, cm³, liters, in³, and ft³.

Torus Volume Calculator

Compute the volume of a donut-shaped solid (torus) from the major and minor radii or from the inner and outer radius. Supports metric and imperial units and shows results in multiple volume units.

Torus volume – interactive calculator

Input mode

Major radius R and minor radius r Inner radius R_in and outer radius R_out

Use R and r if you already know the center-to-center radius and tube radius. Use inner/outer radius if you can measure the hole and outer edge.

Length unit

Major radius R

Distance from the center of the hole to the center of the tube, in the selected length unit.

Minor radius r

Radius of the tube (cross-section circle), in the same unit as R. For a proper ring torus, R > r > 0.

Results

Quantity	Value	Unit

Note: results are rounded to a reasonable number of significant digits for readability. For engineering or scientific work, keep sufficient precision in your input data.

Torus volume formula

A torus (commonly described as a donut-shaped solid) can be obtained by rotating a circle of radius r around an axis in the same plane, at distance R from the center of the circle. Here:

R – major radius: distance from the center of the torus to the center of the tube;
r – minor radius: radius of the tube (the cross-section circle).

Standard ring torus volume

Using Pappus's centroid theorem or direct integration, the volume of a ring torus is:

\[ V = 2 \pi^2 R r^2 \]

This formula assumes a ring torus, where \(R > r > 0\). If \(R \leq r\), the torus becomes horn-shaped or self-intersecting, and the standard formula is not appropriate for most practical applications.

Volume from inner and outer radius

In practice you may measure the torus from its inner radius \(R_{\text{in}}\) (hole radius) and outer radius \(R_{\text{out}}\) (overall radius). These are related to \(R\) and \(r\) by:

\[ R = \frac{R_{\text{out}} + R_{\text{in}}}{2}, \qquad r = \frac{R_{\text{out}} - R_{\text{in}}}{2} \] \[ V = 2 \pi^2 R r^2 \]

The calculator automatically performs these conversions when you choose the “inner/outer radius” mode, ensuring consistent units throughout.

Worked example

Suppose you have a torus with:

major radius \(R = 0.5 \, \text{m}\)
minor radius \(r = 0.1 \, \text{m}\)

The volume is:

\[ V = 2 \pi^2 R r^2 = 2 \pi^2 \cdot 0.5 \cdot (0.1)^2 \approx 0.0987 \, \text{m}^3 \] which corresponds to about \(98.7 \, \text{liters}\).

Assumptions & limitations

The torus is perfectly rotationally symmetric and generated by a circle.
All radii are measured relative to the same center line and use the same unit.
The calculator works with standard ring tori where \(R > r\). Degenerate or self-intersecting shapes are not supported.