Math Playground
Geometry

Torus

A donut. One hole, but a surprising amount of geometry.

A torus is a donut. Take a circle, then sweep its center around a bigger circle — what you trace out is a torus. It's the simplest shape in 3D space that has a hole going through it.

Volume
V = 2 π² R r²

R = distance from torus center to tube center; r = tube radius.

Surface area
A = 4 π² R r

Topologically, a coffee mug and a donut are the same shape — both have exactly one hole. That's the famous 'topologists can't tell their breakfast apart' joke.