Torus Revolution Volume Calculator

Torus volume of revolution

Take a circle and spin it around an axis that sits in its plane but doesn't touch it, and you trace out a torus (picture a doughnut or a tyre). Its volume comes to V = 2π²·R·r², where R is the distance from the axis of revolution to the centre of the tube and r is the radius of the tube. Example: with R = 6 cm and r = 2 cm you get V = 2π²·6·4 ≈ 473.7 cm³. Where does the formula come from? Pappus's second theorem: the volume of a solid of revolution is the area of the generating region (πr²) multiplied by the distance its centroid travels (2πR).

Applications

Automotive engineers go from tyre volume to rubber mass with V·ρ. In plasma physics, the ITER tokamak fusion reactor uses a toroidal plasma chamber with R ≈ 6.2 m and r ≈ 2 m. Mechanical engineering has O-rings, bearings and donut seals; jewellers make rings; architects design toroidal arches; and bakers estimate batter for doughnuts and ring cakes.

FAQ

What is the difference between R and r? Think of R as the "doughnut radius", measured from the axis to the centre of the tube, and r as the "tube radius". Add them up for the total outer radius (R+r); the hole in the middle has radius R−r.

Does R need to be greater than r? If you want a "ring" torus with an actual hole in the middle, then yes, R > r. When R = r the hole pinches shut (a horn torus), and when R < r the surface starts crossing itself (a spindle torus), at which point the formula stops describing a simple physical volume.

Why π²? The two factors of π come from two different circles. One is the circular cross-section area (πr²), the other is the circular path the centroid sweeps out (2πR). Multiply them and you land on 2π²·R·r².