An implicit equation in Cartesian coordinates for a torus radially symmetric about the z-axis is

or the solution of f(x, y, z) = 0, where

Algebraically eliminating the square root gives a quartic equation,

The three classes of standard tori correspond to the three possible aspect ratios between R and r:

When R > r, the surface will be the familiar ring torus or anchor ring. R = r corresponds to the horn torus, which in effect is a torus with no "hole". R < r describes the self-intersecting spindle torus; its inner shell is a lemon and its outer shell is an apple
When R = 0, the torus degenerates to the sphere.When R ≥ r, the interior

of this torus is diffeomorphic (and, hence, homeomorphic) to a product of a Euclidean open disk and a circle. The volume of this solid torus and the surface area of its torus are easily computed using Pappuss centroid theorem, giving:
These formulas are the same as for a cylinder of length 2πR and radius r, obtained from cutting the tube along the plane of a small circle, and unrolling it by straightening out (rectifying) the line running around the center of the tube. The losses in surface area and volume on the inner side of the tube exactly cancel out the gains on the outer side.
The ring torus is one way to embed this space into Euclidean space, but another way to do this is the Cartesian product of the embedding of S1 in the plane with itself. This produces a geometric object called the Clifford torus, a surface in 4-space. In the field of topology, a torus is any topological space that is homeomorphic to a torus. The surface of a coffee cup and a doughnut are both topological tori with genus one. An example of a torus can be constructed by taking a rectangular strip of flexible material, for example, a rubber sheet, and joining the top edge to the bottom edge, and the left edge to the right edge, without any half-twists (compare Möbius strip). Geometry
A torus can be defined parametrically by:
where

θ, φ are angles which make a full circle, so their values start and end at the same point,
R is the distance from the center of the tube to the center of the torus,
r is the radius of the tube.Angle θ represents rotation around the tube, whereas φ represents rotation around the torus axis of revolution. R is known as the "major radius" and r is known as the "minor radius". The ratio R divided by r is known as the "aspect ratio". The typical doughnut confectionery has an aspect ratio of about 3 to 2.