Torus Volume & Surface

Volume and surface area of a torus

Take a circle of radius r (call it the "tube") and spin it around an axis sitting a distance R from the tube's centre, with R > r. The surface you sweep out is a torus. Both its volume and its surface area drop right out of Pappus's centroid theorems: V = 2π²·R·r² and S = 4π²·R·r. In words, you take the area or the circumference of the disc and multiply by 2πR, the distance the centroid travels as it goes around. Plug in R = 10 and r = 3 and you get V = 2π²·10·9 = 180π² ≈ 1776.53, with S = 4π²·10·3 = 120π² ≈ 1184.35.

Applications

You run into the shape constantly. Tyres, donuts and life buoys are the obvious ones. Less obvious: the chambers of tokamak fusion reactors like ITER and JET, where magnetic confinement comes out toroidal as a matter of course, and the toroidal inductor coils used in EMC filters, where a closed magnetic circuit keeps stray fields very low. Ring-shaped seals are another. And in topology the torus is the standard genus-1 surface, which is just a sphere with one handle attached.

FAQ

What is the difference between R and r? R is the major radius, measured from the centre of the torus out to the centre of the tube. r is the minor radius, which is just the radius of the tube itself.

What if R = r? Then you have a horn torus, where the central hole shrinks down to a single point. Push it further so that R < r and you end up with a spindle torus: the surface crosses itself, and at that stage the formulas above stop giving you the real volume of the solid region.

Why are inductor cores toroidal? The magnetic flux stays in a closed loop inside the ring, so very little electromagnetic interference leaks out. That matters in audio gear, RF circuits and switching power supplies.