Inmathematics, atoroid is asurface of revolution with a hole in the middle. 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 acircle, then the object is called atorus.

The termtoroid is also used to describe atoroidal polyhedron. In this context a toroid need not be circular and may have any number of holes. Ag-holedtoroid can be seen as approximating the surface of atorus having atopologicalgenus,g, of 1 or greater. TheEuler characteristic χ of ag holed toroid is 2(1-g).^[2]

The torus is an example of a toroid, which is the surface of adoughnut. Doughnuts are an example of asolid torus created by rotating a disk, and are not toroids.

Toroidal structures occur in both natural and synthetic materials.^[3]

A toroid is specified by the radius of revolutionR measured from the center of the section rotated. For symmetrical sections volume and surface of the body may be computed (with circumferenceC and areaA of the section):

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.

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

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

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.

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

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

Pappus's centroid theorem generalizes the formulas here to arbitrary surfaces of revolution.

• Toroidal inductors and transformers
• Toroidal propeller
• Annulus
• Solenoid
• Helix

• The dictionary definition oftoroid at Wiktionary

• Topology
• Geometric shapes

• Articles with short description
• Short description is different from Wikidata
• Use dmy dates from August 2024