Torus Volume Calculator
From a major radius R and a minor radius r, get the volume and surface area of a ring torus — the donut shape behind O-rings, tyres, and tube-shaped tanks.
Two radii, two answers
Enter the major radius R (centre of the hole to the centre of the tube) and the minor radius r (the tube) and the calculator returns the volume (2π²Rr²) and the surface area (4π²Rr) at once.
Keep units consistent
The radii are unit-agnostic — your answers come back in the same unit (squared for the surface, cubed for the volume), so don't mix centimetres with inches.
What is a torus volume calculator?
Two radii in, full donut out
A torus volume calculator turns two measurements — the major radius R and the minor radius r — into the numbers that describe a whole ring torus: how much it holds (volume) and the area of its outer skin (surface area). A torus is the donut or ring shape you get by spinning a circle around an axis it does not cross. Both outputs are fixed once you know the two radii, because every torus shares the same constant π (pi). That makes those two inputs all you need for O-rings, tyres, inner tubes, and tube-shaped tanks.
Enter the major and minor radii in any length unit to get the volume and surface area of the donut instantly.
Two short formulas, both built from the major radius R, the minor radius r, and the constant π (about 3.14159).
volume = 2 × π² × R × r²The surface area — the outer skin of the ring — is 4 × π² × R × r. Both come from picturing the torus as a cylinder bent into a loop: the tube (cross-section area πr², wall circumference 2πr) sweeps once around a circle of circumference 2πR. The volume is then 2πR × πr² = 2π²Rr², and the surface area is 2πR × 2πr = 4π²Rr.
Suppose you have a torus with a major radius of 10 and a minor radius of 3.
Surface area
4 × π² × 10 × 3 = 1184.352528 square units — the outer skin of the ring.
Volume
2 × π² × 10 × 3² = 1776.528792 cubic units — the space inside the donut.
The two outputs answer two everyday questions. The volume (about 1776.528792 cubic units for R = 10, r = 3) is how much the torus holds — the air in an inner tube, the fluid in a ring-shaped tank, the rubber in a tyre. The single most useful picture is that a torus is a cylinder of length 2πR — the circle the tube sweeps — wrapped into a ring: bigger R stretches the ring wider, while bigger r fattens the tube. The surface area (about 1184.352528 square units) is the outer skin you would coat, paint, or seal, useful for material estimates on O-rings and gaskets. Both scale with R, but the volume grows with the square of r while the surface grows only linearly, so a fatter tube adds volume far faster than skin. π is the thread tying it together — the same constant links the two radii to both the volume and the surface of every torus, large or small.
The formulas are exact, but a couple of practical points are worth keeping in mind.
Ring torus and consistent units
These formulas describe a ring torus — a circular tube whose major radius R is at least as large as its minor radius r, so the tube does not overlap itself through the centre. A horn torus (R = r) sits at the boundary, and a spindle torus (R less than r, where the tube passes through the centre) is not covered. The radii are also unit-agnostic, so the answers are only meaningful if you keep one unit throughout: radii in centimetres give a volume in cubic centimetres and a surface in square centimetres, never a mix.