# 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):

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

Which gives for square and circular sections:

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

${\displaystyle V=a^{2}2\pi R}$
${\displaystyle S=8a\pi R}$

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

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