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In mathematics, a **toroid** is a surface of revolution with a hole in the middle. 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]}

The torus is an example of a toroid, which is the surface of a doughnut. Doughnuts are an example of a solid torus created by rotating a disk, and should not be confused with toroids.

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):

The volume (V) and surface area (S) of a toroid are given by the following equations, where A is the area of the square section of side, and R is the radius of revolution.

The volume (V) and surface area (S) of a toroid are given by the following equations, where r is the radius of the circular section, and R is the radius of the overall shape.

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

- The dictionary definition of
*toroid*at Wiktionary