Toroid
Surface of revolution with a hole in the middle
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. 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).
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 are not toroids.
Toroidal structures occur in both natural and synthetic materials.
01Equations
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.
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.
Pappus's centroid theorem generalizes the formulas here to arbitrary surfaces of revolution.
Sources and credits
This article is adapted from the Wikipedia article “Toroid”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.
Images, from Wikimedia Commons:
- Torus.png by Lucas Vieira, Public domain
- Toroid by Zureks.svg by Zureks, Public domain
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