represents a torus centered at {x,y,z} with inner radius rinner and outer radius router.


  • Torus is also known as torus of revolution.
  • Torus can be used as a geometric region and a 3D graphics primitive.
  • Torus[] is equivalent to Torus[{0,0,0},{1/2,1}].
  • Torus represent the shell .
  • Torus can be used in Graphics3D.
  • Graphics rendering is affected by directives such as FaceForm, Specularity, Opacity and color.


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Basic Examples  (2)

A standard torus at the origin:

Area and centroid:

Scope  (18)

Graphics  (9)

Specification  (4)

The standard torus:

Filled tori with different outer radii:

Filled tori with different inner radii:

Short form for a torus with radii at the origin:

Styling  (4)

Colored tori:

Different properties can be specified for the front and back of faces using FaceForm:

Filled tori with different specular exponents:

White torus that glows red:

Opacity specifies the face opacity:

Coordinates  (1)

Points can be Dynamic:

Regions  (9)

Embedding dimension is the dimension of the space in which the torus lives:

Geometric dimension is the dimension of the shape itself:

Membership testing:

Get conditions for point membership:



Distance from a point:

The equidistance contours for a torus:

Signed distance from a point:

Nearest point in the region:

Nearest points to an enclosing sphere:

A torus is bounded:

Find its range:

Optimize over a torus region:

Solve equations in a torus region:

Applications  (2)


Use Torus to render nodes in a GraphPlot3D:

Properties & Relations  (3)

An implicit specification of a torus generated by ContourPlot3D:

A parametric specification of a torus generated by ParametricPlot3D:

Torus is the RegionBoundary of FilledTorus:

Neat Examples  (3)

Random torus collections:

Double helix:

Nested tori:

Wolfram Research (13), Torus, Wolfram Language function,


Wolfram Research (13), Torus, Wolfram Language function,


Wolfram Language. 13. "Torus." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (13). Torus. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2021_torus, author="Wolfram Research", title="{Torus}", year="13", howpublished="\url{}", note=[Accessed: 24-January-2022 ]}


@online{reference.wolfram_2021_torus, organization={Wolfram Research}, title={Torus}, year={13}, url={}, note=[Accessed: 24-January-2022 ]}