Torus Revolution Surface Calculator
Computes torus revolution surface area from major radius R and tube radius r using the 4 pi squared times R r formula.
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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.
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The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.