Torus Surface Area Calculator
From a major radius and a minor radius, get the surface area and the volume — the two numbers that describe any ring torus.
Two inputs, two answers
Enter the major radius R and the minor radius r and the calculator returns the surface area (4π²Rr) and the volume (2π²Rr²) at once.
R must exceed r
A ring torus needs the major radius larger than the minor radius — otherwise the tube would overlap itself and the formulas no longer describe a donut.
What is a torus surface area calculator?
Two radii in, full donut out
A torus surface area calculator turns two measurements — the major radius R (centre of the donut to the centre of the tube) and the minor radius r (the tube's own radius) — into the numbers that describe a whole ring torus: the area of its outer skin (surface area) and how much it holds (volume). Each one is fixed once you know R and r, because every torus shares the same constant π (pi). That makes those two inputs all you need for O-rings, inner tubes, donuts, life rings, and any geometry problem where a ring shape shows up.
Enter the major and minor radius in any length unit to get the surface area and volume instantly.
Two short formulas, both built from the major radius R, the minor radius r, and the constant π (about 3.14159).
surface = 4 × π² × R × rThe surface area is the outer skin of the ring — 4 × π² × R × r. The volume, the space inside the tube, is 2 × π² × R × r². Both come from Pappus's theorem: a torus is a circle of radius r swept around an axis at distance R, so its surface is the tube's circumference (2πr) times the path of its centre (2πR), and its volume is the tube's area (πr²) times that same path.
Suppose you have a torus with a major radius of 5 and a minor radius of 2.
Volume
2 × π² × 5 × 2² = 2 × π² × 5 × 4 = 394.784176 cubic units — the space inside the tube.
Surface area
4 × π² × 5 × 2 = 394.784176 square units — the outer skin (the two answers coincide only for these particular numbers).
The two outputs answer two different everyday questions. The surface area (about 394.784176 square units for R = 5, r = 2) is the outer skin of the ring — the area you would paint, plate, or wrap, useful for coating an O-ring or estimating the icing on a donut. The volume (also about 394.784176 here, a coincidence of these inputs) is how much the tube holds — the rubber in an inner tube, the air in a life ring, the dough in a donut. The single most useful insight is that both numbers scale with the product R × r: double either radius and the surface doubles too, while the volume grows with R × r², so the tube thickness counts twice. π appears squared in both formulas because a torus is built from two circles — one swept around the other — and each contributes its own π. Keep R well above r and you have a clean ring; let r approach R and the hole shrinks toward a "horn torus" where the inner edge just touches.
The formulas are exact, but a couple of practical points are worth keeping in mind.
Ring tori and consistent units
These formulas describe a ring torus — a circular tube whose major radius R is strictly larger than its minor radius r, so the hole in the middle stays open. If r equals or exceeds R the tube overlaps itself (a horn or spindle torus) and the surface and volume formulas no longer apply, so the calculator returns nothing. The two radii are also unit-agnostic, so the answers are only meaningful if you keep one unit throughout: radii in centimetres give a surface in square centimetres and a volume in cubic centimetres, never a mix.