A torus is the doughnut shape: a circular tube bent into a closed ring. This torus volume calculator returns its volume and surface area from the two radii that define it, or from the inner and outer diameters if that is how your part is specified. It also reports the outer diameter, the hole diameter and the length of the tube's centre line, which is the number you need if you are bending stock into a ring.
Arb Digital built this page with a hard validation rule that most torus pages skip: if the minor radius is equal to or greater than the major radius, the tool refuses to return a volume and explains why. That case does not describe a ring at all, and quietly printing a number for it is how bad answers spread.
What This Torus Volume Calculator Does
A ring torus is fully described by two numbers. The major radius R runs from the centre of the hole to the centre of the tube. The minor radius r is the radius of the tube's circular cross section. Everything else follows from those two.
The calculator produces the volume of solid material in the ring, the total surface area of the outside, the overall outer diameter 2(R + r), the diameter of the central hole 2(R − r), and the centre-line circumference 2πR, which is the length of tube you would need to bend to form the ring. Volume is also reported in litres and US gallons where the size makes that meaningful.
In the second input mode you enter the outer and inner diameters instead, which is how O-rings, gaskets, washers, tyres, pipe collars and lifebuoys are almost always dimensioned in catalogues. The calculator converts: R is the average of the two radii, and r is half their difference. That step alone removes a lot of hand arithmetic and a lot of mistakes.
How to Use It
- Choose your input style. Use the two radii for textbook problems, or inner and outer diameters for real parts.
- Enter the two values in a single consistent unit. The field labels change with the mode so there is no ambiguity.
- Pick your unit. Areas are returned squared and volume cubed, with capacity underneath.
- Check the validation message. If the tube radius is too large for the ring radius, the calculator tells you instead of printing a meaningless figure.
- Read the working line to see the substitution with your own numbers.
The Formula: How a Torus Is Calculated
With major radius R and minor radius r:
- Volume: V = 2π²Rr². With R = 10 cm and r = 3 cm: 2 × π² × 10 × 9 = 180π² = 1,776.53 cm³, or about 1.78 litres.
- Surface area: A = 4π²Rr. Here 4 × π² × 10 × 3 = 120π² = 1,184.35 cm².
- Outer diameter: 2(R + r) = 26 cm.
- Hole diameter: 2(R − r) = 14 cm.
- Centre-line length: 2πR = 62.83 cm.
Going the other way, if you know the outer diameter D and the inner diameter d, then r = (D − d) ÷ 4 and R = (D + d) ÷ 4. A 26 cm outer and 14 cm inner gives r = 3 and R = 10, recovering the same ring. The complete set of torus identities is documented at Wolfram MathWorld's torus entry.
Pappus's Theorem: Why the Formula Looks Like That
The two π symbols in 2π²Rr² are not a coincidence, and the formula is easier to remember once you see where it comes from. Pappus's centroid theorem says that the volume of a solid formed by revolving a flat shape around an external axis equals the area of that shape multiplied by the distance its centroid travels.
A torus is a circle of radius r revolved around an axis a distance R away. The circle's area is πr². Its centroid travels a full circle of circumference 2πR. Multiply them: πr² × 2πR = 2π²Rr². The surface area follows the same logic with the circle's perimeter 2πr instead of its area, giving 2πr × 2πR = 4π²Rr.
This has a useful practical consequence. A torus has exactly the same volume as a straight cylinder of radius r and length 2πR, the length of its own centre line. Bending a tube into a ring does not change the amount of material in it, which is exactly what you would expect physically and is reassuring to see fall out of the maths. You can verify that equivalence with the cylinder volume calculator by entering radius r and height 2πR.
Why the Minor Radius Must Be Smaller Than the Major Radius
The formulas above describe a ring torus, the ordinary doughnut with a visible hole through the middle. That shape exists only when r is strictly less than R. Three cases matter:
- r < R: a ring torus. There is a genuine hole of diameter 2(R − r), and the formulas apply.
- r = R: a horn torus. The hole closes to a single point. The surface touches itself at the centre and the shape is no longer a simple ring.
- r > R: a spindle torus. The revolved circle crosses the axis, so the surface passes through itself and the solid overlaps its own interior. The expression 2π²Rr² still returns a number, but it double-counts the overlapping region and does not describe the volume of any physical object.
This is why the calculator refuses to proceed when r is greater than or equal to R rather than printing a figure. If you arrived at that condition from an inner and outer diameter, the usual cause is entering the diameters the wrong way round, or entering a radius in one field and a diameter in the other. In the diameter mode, an inner diameter of zero or less than zero produces exactly this situation, because it means the hole has closed.
Real Parts: O-Rings, Washers and Tyres
Sealing rings are catalogued by inside diameter and cross-section thickness, which map directly onto this calculator: the cross-section thickness is the tube diameter, so r is half of it, and R is the inside radius plus r. A ring listed as 14 mm inside diameter with a 6 mm cross section has r = 3 mm and R = 7 + 3 = 10 mm.
Once you have the volume, material quantity follows from density: mass equals volume times density, so a rubber ring of 1,776.53 cubic centimetres at a density of 1.2 grams per cubic centimetre would weigh about 2.13 kilograms. The density converter handles the unit side of that step when your density figure is in different units from your volume.
The surface area figure matters for coating, plating and heat transfer, and the centre-line length matters for fabrication, since it tells you how much straight stock to cut before bending. A torus is one of the few shapes where the manufacturing dimension and the geometric dimension differ enough that people routinely order the wrong length of material.
Capacity, Units and What This Tool Does Not Do
Because a cubic centimetre is exactly one millilitre, dividing cubic centimetres by 1,000 gives litres, and dividing litres by 3.785411784 gives US gallons. That matters for inflatable rings, hollow toroidal tanks and pipe loops. Note that this calculator gives the volume of the solid torus. A hollow toroidal tube with a wall thickness holds less than that; to find the internal capacity, compute a second torus using the internal minor radius and subtract if you need the wall material volume.
Cubed unit conversions follow the usual rule, with the linear factor cubed: one cubic metre is 1,000,000 cubic centimetres, per NIST's guide to the SI. If you already have a volume and simply want it in other units, the volume converter is the right tool; this page is for computing a volume from physical dimensions. For the flat circular cross section on its own, use the circle calculator.
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Browse All Free Tools Talk To Arb DigitalCommon Mistakes to Avoid
- Swapping R and r. Volume depends on r squared but only on R to the first power, so swapping them changes the answer dramatically.
- Entering a diameter in a radius field. Doubling r multiplies the volume by four. Use the diameter input mode instead of halving by hand.
- Using the formula when r is greater than or equal to R. That is a spindle or horn torus, and 2π²Rr² double-counts the overlap.
- Measuring R from the outside edge. The major radius runs to the centre of the tube, not to its outer or inner surface.
- Confusing solid volume with internal capacity. A hollow ring holds less than the solid volume of a ring with the same outer dimensions.
Related Free Tools From Arb Digital
The cylinder volume calculator gives the straight-tube equivalent that Pappus's theorem says has identical volume. The sphere volume calculator covers the other classic revolved solid, and the circle calculator handles the cross section on its own. For unit work, the volume converter and length converter do the conversions. Browse the full free online tools hub for everything else.
Frequently Asked Questions
Volume is 2π²Rr², where R is the major radius from the centre of the hole to the centre of the tube and r is the tube radius. With R = 10 cm and r = 3 cm the volume is 1,776.53 cubic centimetres.
Surface area is 4π²Rr. For a torus with a 10 cm major radius and a 3 cm minor radius that comes to 1,184.35 square centimetres.
Because only then does the shape have a real hole. If r equals R the hole closes to a point, and if r is larger the surface passes through itself, so the standard formula double-counts the overlapping region.
The minor radius is the outer diameter minus the inner diameter, divided by four. The major radius is the outer plus the inner diameter, divided by four. A 26 cm outer and 14 cm inner gives r = 3 cm and R = 10 cm.
Pappus's centroid theorem says the volume equals the cross-section area times the distance the centroid travels. That distance is the centre-line circumference 2πR, so the ring matches a cylinder of radius r and that length.
Multiply the volume by the material density in matching units. A volume of 1,776.53 cubic centimetres at 1.2 grams per cubic centimetre gives about 2.13 kilograms.
No. It returns the volume of a solid torus. A hollow toroidal tube with wall thickness holds less, so compute a second torus using the internal tube radius to find the internal capacity.
Results are mathematical values rounded for display. Manufactured rings carry tolerance, compression set and surface finish variation, so treat these figures as nominal.
