Torus Surface Area Calculator (Donut)

A torus is a donut shape — defined 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”).

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’s the glaze coverage area for one donut. A 16-oz tub of glaze covers about 30-50 donuts depending on glaze viscosity and donut size.

Worked example — bicycle inner tube surface: A road bike inner tube with R = 33 cm, r = 1 cm (inflated for a 25 mm tire). SA = 4π² × 33 × 1 ≈ 1,302 cm² = 0.13 m² of rubber surface.

If you wanted to apply tire sealant (Stan’s, Slime) coating, that’s the inside area to coat. Real sealants only cover the lower portion in normal use, but if you wanted full coverage, this is what you’d need.

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 (the donut-shape works well 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.

This trick works for cones (rotating triangle), spheres (rotating semicircle), cylinders (rotating rectangle), and any other rotation surface. It’s one of the most useful theorems in classical geometry — invented by Pappus of Alexandria around 320 CE, rediscovered 1,500 years later by Paul Guldin.

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).