What is a torus?
A doughnut shape: a circle of tube swept around a central axis. It is described by two radii, R from the centre to the tube's centre, and r the tube's own radius.
// maths › 3D Solids
Torus Volume Calculator
Find the volume and surface area of a torus (doughnut) from R, the distance from the centre to the tube centre, and r, the tube radius.
volume = 2π² R r² ; surface area = 4π² R r
Frequently asked questions
How do I find its volume?
Use 2 pi squared times R times r squared. It is the same as taking a cylinder of length 2 pi R (the path the tube travels) and tube radius r, then bending it into a ring.
What is the surface area?
4 pi squared times R times r. Like the volume, it comes from the tube's circumference times the distance its centre travels around the ring.
Why must r be smaller than R?
So the doughnut has a hole. If r reaches or exceeds R, the tube overlaps itself in the middle and the simple formulas no longer describe a clean ring, which is why the calculator checks it.
Where do toruses appear?
Doughnuts and bagels, inner tubes and O-rings, lifebuoys, and the magnetic confinement chambers of fusion reactors.