Torus Volume Calculator
Torus volume calculator for donut-shaped solids. Compute volume from major/minor radii or inner/outer radius, with 2π²Rr² formula, unit conversions, examples, and FAQ.
Full original guide (expanded)
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
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.
Distance from the center of the hole to the center of the tube, in the selected length unit.
Radius of the tube (cross-section circle), in the same unit as R. For a proper ring torus, R > r > 0.
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:
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:
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.
Frequently asked questions
Formula (LaTeX) + variables + units
V = 2 \pi^2 R r^2
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 = 2 \pi^2 \cdot 0.5 \cdot (0.1)^2 \approx 0.0987 \, \text{m}^3
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Standard ring torus volume Using Pappus's centroid theorem or direct integration, the volume of a ring torus is: \[ V = 2 \pi^2 R r^2 \]
\[ R = \frac{R_{\text{out}} + R_{\text{in}}}{2}, \qquad r = \frac{R_{\text{out}} - R_{\text{in}}}{2} \] \[ V = 2 \pi^2 R r^2 \]
\[ V = 2 \pi^2 R r^2 = 2 \pi^2 \cdot 0.5 \cdot (0.1)^2 \approx 0.0987 \, \text{m}^3 \] which corresponds to about \(98.7 \, \text{liters}\).
- No variables provided in audit spec.
- NIST — Weights and measures — nist.gov · Accessed 2026-01-19
https://www.nist.gov/pml/weights-and-measures - FTC — Consumer advice — consumer.ftc.gov · Accessed 2026-01-19
https://consumer.ftc.gov/
Last code update: 2026-01-19
- Initial audit spec draft generated from HTML extraction (review required).
- Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
- Confirm sources are authoritative and relevant to the calculator methodology.