Torus Volume Calculator

The formula

V = 2π²Rr², A = 4π²Rr

Pappus’s theorem does the work

A torus is a circle swept around an axis, so its volume is simply the tube’s cross-sectional area times the distance its centre travels: πr² × 2πR. The surface area follows the same logic with the tube’s circumference: 2πr × 2πR. That is Pappus’s centroid theorem, and it turns an awkward-looking solid into two multiplications.

Volume scales with the cube

Double every length and the surface area quadruples while the volume grows eightfold. This is why doubling a container’s dimensions gives eight times the capacity but only four times the material — and why scaling models up rarely works the way intuition suggests.

The one-third family

Cones and pyramids are always exactly one third of the prism or cylinder that would enclose them, whatever the base shape. A sphere sits at two thirds of its enclosing cylinder — the result Archimedes asked to have carved on his tomb.

Units

Enter every dimension in the same unit; volume comes out cubed and surface area squared. If you work in centimetres, one litre is 1,000 cm³, which is why the capacity row is shown on the box and cylinder pages.