# Toroid

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{{Short description|Surface of revolution with a hole in the middle}}
{{Other uses}}
{{Use dmy dates|date=August 2024}}
thumb|A toroid using a square.
[[File:Torus.png|thumb|A [torus](/source/torus) is a type of toroid.]]

In [mathematics](/source/mathematics), a '''toroid''' is a [surface of revolution](/source/surface_of_revolution) with a hole in the middle. The [axis of revolution](/source/axis_of_revolution) passes through the hole and so does not intersect the surface.<ref>{{MathWorld|Toroid|Toroid}}</ref> 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](/source/circle), then the object is called a ''[torus](/source/torus)''.

The term ''toroid'' is also used to describe a [toroidal polyhedron](/source/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](/source/torus) having a [topological](/source/Topology_(Mathematics)) [genus](/source/Genus_(mathematics)), ''g'', of 1 or greater. The [Euler characteristic](/source/Euler_characteristic) &chi; of a ''g'' holed toroid is 2(1−''g'').<ref>Stewart, B.; "Adventures Among the Toroids:A Study of Orientable Polyhedra with Regular Faces", 2nd Edition, Stewart (1980).</ref>

The torus is an example of a toroid, which is the surface of a [doughnut](/source/doughnut). Doughnuts are an example of a [solid torus](/source/solid_torus) created by rotating a disk, and are not toroids.

Toroidal structures occur in both natural and synthetic materials.<ref>{{Cite journal |last=Carroll |first=Gregory T. |last2=Jongejan |first2=Mahthild G. M. |last3=Pijper |first3=Dirk |last4=Feringa |first4=Ben L. |date=2010 |title=Spontaneous generation and patterning of chiral polymeric surface toroids |url=https://xlink.rsc.org/?DOI=c0sc00159g |journal=Chemical Science |language=en |volume=1 |issue=4 |pages=469 |doi=10.1039/c0sc00159g |issn=2041-6520}}</ref>

== 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.

:<math>V = 2 \pi R A</math>
:<math>S = 2 \pi R C</math>

=== 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.

:<math>V = 2  \pi^2 r^2 R</math>
:<math>S = 4 \pi^2 r R</math>

[Pappus's centroid theorem](/source/Pappus's_centroid_theorem) generalizes the formulas here to arbitrary surfaces of revolution.

== See also ==
*[Toroidal inductors and transformers](/source/Toroidal_inductors_and_transformers)
*[Toroidal propeller](/source/Toroidal_propeller)
*[Annulus](/source/Annulus_(mathematics))
*[Solenoid](/source/Solenoid)
*[Helix](/source/Helix)

==Notes==
{{Reflist}}

==External links==
*{{Wiktionary-inline|toroid}}

{{Authority control}}

Category:Surfaces of revolution

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Adapted from the Wikipedia article [Toroid](https://en.wikipedia.org/wiki/Toroid) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Toroid?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.