# Torus

## Torus

Solid generated by the rotation of a circle C of radius r about an axis in the same plane at a distance R from its centre.

### Properties

The shape of a torus is determined by the two real parameters R and r:

• If R = 0, the corresponding torus is a sphere; in this case, it is also a solid torus
• If R < r, the corresponding torus is a self-intersecting spindle torus, which resembles a pumpkin (the poles are flattened).
• If R = r, the two poles coincide and it is a horn torus
• If R > r, the most common shape of the torus is obtained, that is, a ring torus, which is shaped like a doughnut, as shown in the illustration above.

### Formulas

The volume V of a torus of radius R generated by a circle of radius r is given by:

V = 2$$π^{2}Rr^{2}$$

The area A of a torus of radius R generated by a circle of radius r is given by: A = 4$$π^{2}Rr$$