In mathematics, a toroid is a

surface of revolution A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation. Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on ...

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 A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...

, then the object is called a

torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does n ...

. The term ''toroid'' is also used to describe a

toroidal polyhedron In geometry, a toroidal polyhedron is a polyhedron which is also a toroid (a -holed torus), having a topological genus () of 1 or greater. Notable examples include the Császár and Szilassi polyhedra. Variations in definition Toroidal polyh ...

. 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

having a topological

genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...

, ''g'', of 1 or greater. The

Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological spac ...

χ of a ''g'' holed toroid is 2(1-''g'').Stewart, B.; "Adventures Among the Toroids:A Study of Orientable Polyhedra with Regular Faces", 2nd Edition, Stewart (1980). The torus is an example of a toroid, which is the surface of a

doughnut A doughnut or donut () is a type of food made from leavened fried dough. It is popular in many countries and is prepared in various forms as a sweet snack that can be homemade or purchased in bakeries, supermarkets, food stalls, and fra ...

. Doughnuts are an example of a

solid torus In mathematics, a solid torus is the topological space formed by sweeping a disk around a circle. It is homeomorphic to the Cartesian product S^1 \times D^2 of the disk and the circle, endowed with the product topology. A standard way to visual ...

created by rotating a disk, and should not be confused with toroids.

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. :

V = 2 \pi R A

S = 2 \pi R C

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. :

V = 2  \pi^2 r^2 R

S = 4 \pi^2 r R

Notes

External links

* Topology Geometric shapes {{geometry-stub

Equations

Square Toroid

Circular Toroid

See also

Notes

External links