Torus Revolution Surface Calculator

Surface area of a torus of revolution

Take a circle of radius r (the tube) and spin it around an external axis sitting at distance R from the circle's centre. What you get is a torus of revolution. Pappus's first theorem tells us the surface equals the perimeter of the generating curve times the distance its centroid travels, which works out to S = (2π·r) · (2π·R) = 4π²·R·r. Run the numbers on a Michelin tyre: rim diameter 22" puts R at roughly 0.41 m out to the tread centre, and with a tread tube radius near 0.075 m you land on S = 4π²·0.41·0.075 ≈ 1.21 m² per tyre. A car's four tyres bring that to about 4.9 m² of rubber. Want to coat ten tyres for storage? You'd be looking at paint for somewhere around 12 m².

Applications

Tyre engineering leans on it, since rolling resistance and the contact patch both come down to surface and tube geometry. So do toroidal inductor cores, where surface times cross-section fixes the effective magnetic area. You also see it in gasket and O-ring manufacturing, in estimating the wall area of a tokamak for first-wall coating on ITER, and in ring-shaped architectural roofs. Apple Park HQ, for one, is almost a torus.

FAQ

Why does Pappus's theorem work? Each infinitesimal arc of length ds on the generating circle sweeps out a strip of area ds·(2π·ρ), where ρ is how far that arc sits from the axis. Integrate over the whole curve and you're left with perimeter × (2π × centroid distance). For a circle that comes to 2πr · 2πR.

What if R < r? Now the generating circle crosses the axis and the surface folds in on itself, giving you a spindle torus. You can still plug into 4π²·R·r and get a number out, but it stops matching the area of the outer surface you actually see.

How does this differ from volume? Volume falls out of Pappus's second theorem instead: take the area of the generating region times the distance its centroid travels, so V = π·r² · 2π·R = 2π²·R·r².