# Toroid

In mathematics, a **toroid** is a surface of revolution with a hole in the middle, like a doughnut, forming a solid body. The axis of revolution passes through the hole and so does not intersect the surface.[1] For example, when a rectangle is rotated around an axis parallel to one of its edges, then a hollow rectangle-section ring is produced. If the revolved figure is a circle, then the object is called a torus.

The term *toroid* is also used to describe a toroidal polyhedron. In this context a toroid need not be circular and may have any number of holes. A *g*-holed *toroid* can be seen as approximating the surface of a torus having a topological genus, *g*, of 1 or greater. The Euler characteristic χ of a *g* holed toroid is 2(1-*g*).[2]

A toroid is specified by the radius of revolution *R* measured from the center of the section rotated. For symmetrical sections volume and surface of the body may be computed (with circumference *C* and area *A* of the section):

Which gives for square and circular sections:

The volume and surface of a toroid with square section of side *a* are given by

The volume and surface of a toroid with circular section of radius *r* (torus) are given by

## Notes

- Weisstein, Eric W. "Toroid".
*MathWorld*. - Stewart, B.; "Adventures Among the Toroids:A Study of Orientable Polyhedra with Regular Faces", 2nd Edition, Stewart (1980).