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

Details and Options


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Basic Examples  (2)

The standard filled torus at the origin:

Volume and centroid:

Scope  (18)

Graphics  (9)

Specification  (4)

The standard filled torus:

Filled tori with different outer radii:

Filled tori with different inner radii:

Short form for a filled torus with radii at the origin:

Styling  (4)

Colored filled tori:

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

Filled tori with different specular exponents:

White filled 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 filled 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 filled torus:

Signed distance from a point:

Nearest point in the region:

Nearest points to an enclosing sphere:

A filled torus is bounded:

Find its range:

Optimize over a filled torus region:

Solve equations in a filled torus region:

Applications  (2)

Find the minimum surface area for a filled torus with volume :

Find the mass of caffeine in a filled torus with a radius of 3 centimeters:

Density of caffeine:

Volume of filled torus:

Mass of caffeine in the filled torus:

Properties & Relations  (1)

Neat Examples  (3)

Random filled torus collections:

Double helix:

Nested filled tori:

Wolfram Research (2021), FilledTorus, Wolfram Language function,


Wolfram Research (2021), FilledTorus, Wolfram Language function,


Wolfram Language. 2021. "FilledTorus." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2021). FilledTorus. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_filledtorus, author="Wolfram Research", title="{FilledTorus}", year="2021", howpublished="\url{}", note=[Accessed: 24-June-2024 ]}


@online{reference.wolfram_2024_filledtorus, organization={Wolfram Research}, title={FilledTorus}, year={2021}, url={}, note=[Accessed: 24-June-2024 ]}