Toroid
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Toroid
A toroid using a square.
A torus is a type of 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]

## Equations

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):

### Square Toroid

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.

${\displaystyle V=2\pi RA}$
${\displaystyle S=2\pi RC}$

### Circular Toroid

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.

${\displaystyle V=2\pi ^{2}r^{2}R}$
${\displaystyle S=4\pi ^{2}rR}$

## Notes

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