As a surface, the torus is topologically S¹×S¹, where S¹ denotes the circle. The n-dimensional torus is (S¹)ⁿ, and such tori are important in the theory of Lie groups. A compact, connected abelian Lie group is a torus.

Also a compact, connected Lie group G will have a maximal torus T. Any two maximal tori are conjugate subgroups, and every element of G lies in some maximal torus.

Suppose that a circle of radius a is rotated through one revolution about a line, in the plane of the circle, a distance b from the centre of the circle, where b > a. The resulting surface or solid is called a torus, the shape of a ‘doughnut’. The surface area of this torus equals 4π‎²ab, and its volume equals 2π‎²a²b. More generally, the term ‘torus’ applies to an orientable closed surface with g holes, called the genus of the torus. See classification theorem for surfaces.