The torus volume calculator computes every key measurement of a perfect ring torus from two inputs — the major radius R and the tube radius r — and also works backwards: supply the volume plus one radius and it solves for the missing dimension. Results include volume, surface area, inner and outer radii, and both characteristic circumferences, all shown with a step-by-step working panel and a labelled cross-section diagram.
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What is a torus?
A torus is the surface (and solid) generated by rotating a circle of radius r about an axis in the same plane that lies at distance R from the circle’s centre. The result looks like a doughnut, an inner tube, or a ring. Two measurements completely define its shape:
- R (major radius): distance from the centre of the torus to the centre of the tube.
- r (minor / tube radius): radius of the circular cross-section of the tube.
The constraint R > r keeps the torus in the standard “ring torus” form where the hole remains open. When R = r the inner hole closes to a point (a “horn torus”), and when R is less than r the torus self-intersects (a “spindle torus”) — this calculator only handles ring tori.
How it works
Two exact closed-form formulas give all the geometry:
Volume: V = 2 x pi^2 x R x r^2
Surface area: A = 4 x pi^2 x R x r
These come from Pappus’s centroid theorem: the volume of a solid of revolution equals the product of the cross-sectional area and the distance travelled by its centroid, and the surface area equals the product of the perimeter and the same travel distance. For a circular cross-section of radius r:
- cross-sectional area = pi x r^2, centroid travel = 2 x pi x R, so V = 2 x pi^2 x R x r^2.
- cross-sectional perimeter = 2 x pi x r, centroid travel = 2 x pi x R, so A = 4 x pi^2 x R x r.
The calculator additionally reports:
- Inner radius: R - r (radius of the central hole).
- Outer radius: R + r (distance from centre to outermost point).
- Tube circumference: 2 x pi x r (circumference of a circular cross-section through the tube).
- Path circumference: 2 x pi x R (length of the centreline of the tube around the full torus).
When the volume is known and one radius is missing, the formulas are simply rearranged:
- Given V and R: r = sqrt(V / (2 x pi^2 x R))
- Given V and r: R = V / (2 x pi^2 x r^2)
All arithmetic uses 64-bit floating-point, giving roughly 15 significant digits.
Worked example
A stainless-steel O-ring has an outer diameter of 50 mm and an inner diameter of 38 mm. What is its volume?
Convert diameters to radii first:
- Outer radius = 50 / 2 = 25 mm
- Inner radius = 38 / 2 = 19 mm
From these: R = (outer + inner) / 2 = (25 + 19) / 2 = 22 mm, r = (outer - inner) / 2 = (25 - 19) / 2 = 3 mm.
Now apply the formulas:
- V = 2 x pi^2 x 22 x 3^2 = 2 x pi^2 x 22 x 9 ≈ 3,947.8 mm^3 (roughly 3.95 cm^3)
- A = 4 x pi^2 x 22 x 3 ≈ 2,631.9 mm^2 (roughly 26.3 cm^2)
| Major R | Tube r | Volume | Surface area |
|---|---|---|---|
| 10 cm | 2 cm | 789.6 cm^3 | 789.6 cm^2 |
| 10 cm | 3 cm | 1,776.5 cm^3 | 1,184.4 cm^2 |
| 10 cm | 5 cm | 4,934.8 cm^3 | 1,973.9 cm^2 |
| 50 mm | 10 mm | 98,696 mm^3 | 19,739.2 mm^2 |
Formula note
Pappus’s centroid theorem (second theorem) states that the volume of a solid of revolution generated by rotating a plane figure through a full 360-degree angle about an external axis equals the product of the area of the figure and the distance travelled by its centroid. For a circle of radius r rotated about an axis at distance R (with R greater than r), the centroid is at the circle’s centre, so the travel distance is 2 x pi x R. Multiplying by the area pi x r^2 gives V = 2 x pi^2 x R x r^2 exactly. The analogous first theorem gives the surface area from the perimeter 2 x pi x r times the same travel: A = 4 x pi^2 x R x r. Both results are exact for all R greater than r, with no approximation involved.
Common uses
Torus geometry appears across engineering and everyday objects far more often than most people realise. O-rings and seals, doughnuts, inner tubes, tyre cross-sections, pipe fittings, toroidal fuel tanks on spacecraft, magnetic confinement chambers in nuclear fusion reactors (tokamaks), coil inductors, and decorative rings are all toroidal. Architects and designers use the torus for arched ceilings, circular handrails and decorative mouldings. In mathematics the torus is a fundamental compact surface used in topology — but for practical calculation of material volumes and surface coatings, this calculator covers all the ordinary engineering cases.