Torus Surface Area Calculator (Donut)

A torus is a donut shape, set by two radii.

• R (major radius): distance from the center of the ring to the center of the tube cross-section.
• r (minor radius): the radius of the circular cross-section, the tube radius.
• For a proper ring torus you need r < R. At r = R the hole closes to a point and the shape is a horn torus; past that the surface passes through itself and the formula below stops describing the outside of anything.

SA = 4 × π² × R × r

Derive this via Pappus’s theorem: the surface area of a solid of revolution equals the perimeter of the rotating shape times the distance traveled by its centroid. For a torus, the rotating shape is a circle of perimeter 2πr; the centroid travels a circle of circumference 2πR. So SA = 2πr × 2πR = 4π²Rr.

Worked example: donut glaze A medium glazed donut has R ≈ 3 cm, r ≈ 1.5 cm. SA = 4π² × 3 × 1.5 = 18π² ≈ 177.7 cm².

That is the coverage area for one donut, and a check on it: the outer diameter comes to 2(R + r) = 9 cm with a 3 cm hole, which is about right for a shop donut. In practice glaze only wets the top and the sides it runs down, so roughly two thirds of this figure is a better estimate than the whole of it.

Worked example: bicycle inner tube surface A road bike inner tube inflated inside a 25 mm tyre on a 700c wheel: R ≈ 33 cm, r ≈ 1 cm. SA = 4π² × 33 × 1 ≈ 1,302 cm² = 0.13 m² of rubber.

That is the figure a manufacturer costs the butyl against, and it explains why tubes are cheap: a tenth of a square metre of thin rubber. It is also the internal area a tyre sealant would have to coat for full coverage, though in normal use sealant only ever pools along the bottom and gets flung around by rotation.

Where torus surface area matters in practice:

• Donut frosting/glaze. Bakery sizing for coating mixtures.
• O-ring coating. Lubricant coverage on rubber O-rings.
• Bicycle inner tube material. Manufacturing cost driven by rubber surface (volume is small).
• Toroidal fuel tanks in aircraft, where the donut shape wraps neatly around landing gear.
• Plasma confinement surface in tokamak fusion reactors, the toroidal wall.
• Lifeboat ring buoys. Material cost for the orange ring.
• Architectural ring features. Concrete or steel toroidal arches and rings.

Pappus’s theorem, the trick:

For ANY surface of revolution, the surface area equals the perimeter of the rotating shape times the distance the centroid travels.

For a torus, the rotating shape is a CIRCLE (perimeter 2πr), centroid traveling distance 2πR. Hence 4π²Rr.

The same trick handles cones (rotate a triangle), spheres (rotate a semicircle), cylinders (rotate a rectangle) and anything else made on a lathe. Pappus of Alexandria stated it around 320 CE. It then sat unused for well over a thousand years until Paul Guldin published the same result in the 1640s, which is why textbooks sometimes call it the Pappus-Guldin theorem.

One catch worth knowing: the centroid has to stay clear of the axis, so the theorem needs r < R here. If the rotating circle crosses the axis, parts of the surface sweep over each other and the product stops counting area once.

Ratio of surface to volume:

For a torus: SA / V = 4π²Rr / (2π²Rr²) = 2/r.

So the surface-to-volume ratio depends only on the minor radius (tube radius), not the major radius. A donut with twice the tube radius has half the surface-to-volume ratio. The ring’s overall size (R) doesn’t affect this.

Sanity check:

• For R = 5, r = 1: SA = 4π² × 5 = 20π² ≈ 197.4 sq units.
• R = 5, r = 2: SA = 4π² × 10 = 40π² ≈ 394.8 sq units. Doubling r doubles SA.
• Doubling R also doubles SA (linear in both).