Ball
✖
Ball
Details and Options

- Ball is also known as center interval, disk, ball, and hyperball.
- Ball can be used as a geometric region and a graphics primitive.
- Ball[] is equivalent to Ball[{0,0,0}].
- Ball[n] for integer n is equivalent to Ball[{0,…,0}], a unit ball in
.
- Ball represents a filled ball
. The region is
dimensional for point p of length
.
- Ball allows p to be any point in
and r any positive real number.
- Ball can be used in Graphics and Graphics3D.
- In graphics, the points p, pi and radii r can be Scaled and Dynamic expressions.
- Graphics rendering is affected by directives such as FaceForm, EdgeForm, Specularity, Opacity, and color.
- Ball[{p1,p2,…},{r1,r2,…}] represents a collection of spheres with centers pi and radii ri.

Background & Context
- Ball is a graphics and geometry primitive that represents a ball in
-dimensional space. In particular, Ball[p,r] represents a (filled-in) ball
in
with center p and radius r, where r may be any non-negative real number and p can have any positive length
. The shorthand form Ball[p] is equivalent to Ball[p,1] and Ball[n] is equivalent to Ball[ConstantArray[0, n],1], while Ball[] autoevaluates to Ball[{0,0,0}].
- Collections of ball objects (multi-balls) of common radius
may be efficiently represented using Ball[{p1,…,pk},r] and balls of varying radii represented using Ball[{p1,…,pk},{r1,…,rk}].
- Ball objects can be visually formatted in two and three dimensions using Graphics and Graphics3D, respectively. The appearance of Ball objects in graphics can be modified by specifying the edge directive EdgeForm (in 2D) or face directive FaceForm (in 3D), color directives such as Red, the transparency and specularity directives Opacity and Specularity, and the style option Antialiasing.
- Ball may also serve as a region specification over which a computation should be performed. For example, Integrate[1,{x,y,z}∈Ball[{0,0,0},r]] and Volume[Ball[{0,0,0},r]] both return the volume
of a 3D ball of radius
.
- Ball is related to a number of other symbols. Sphere represents the boundary of a ball, as can be computed using RegionBoundary[Ball[{x,y,z},r]]. Ball is a generalization of Interval and Disk to arbitrary dimension, and Ellipsoid is a generalization of Ball in the sense that Ball[{p1,…,pk},1] is equivalent to Ellipsoid[{p1,…,pk},ConstantArray[1,k]] for all
. SphericalShell gives a filled shell obtained by removing a small ball from the interior of a larger concentric ball. Ball objects in 3D may be represented as ImplicitRegion[(x-u)2+(y-v)2+(z-w)2≤r2,{u,v,w}] or ParametricRegion[a{Cos[θ]Sin[ϕ],Sin[θ]Sin[ϕ],Cos[ϕ]}-{x,y,z},{{θ,0,2π},{ϕ,0,π},{a,0,r}]. Precomputed properties of the 3D ball and its variants in standard position are available using SolidData["entity", "property"] or EntityValue[Entity["Ball","entity"],"property"], where "entity" is one of "Ball" or "HalfBall".
Examples
open allclose allBasic Examples (2)Summary of the most common use cases
Scope (22)Survey of the scope of standard use cases
Graphics (12)
Specification (4)
The default is a unit ball at the origin in 3D:

https://wolfram.com/xid/0giv1q-ggb1ww

Unit balls in different dimensions:

https://wolfram.com/xid/0giv1q-4hpwzh


https://wolfram.com/xid/0giv1q-snmrge

Balls with different positions and radii:

https://wolfram.com/xid/0giv1q-rnl2wv

Multiple balls with equal radii:

https://wolfram.com/xid/0giv1q-ppozsb

Styling (4)
Balls with different specular exponents:

https://wolfram.com/xid/0giv1q-78t7nc


https://wolfram.com/xid/0giv1q-tdbv9f

Opacity specifies the face opacity:

https://wolfram.com/xid/0giv1q-v6r7w7


https://wolfram.com/xid/0giv1q-q2gss6

Coordinates (4)
Specify coordinates by fractions of the plot range:

https://wolfram.com/xid/0giv1q-z39q1r


https://wolfram.com/xid/0giv1q-nxagkd

Specify radius by fractions of the plot range:

https://wolfram.com/xid/0giv1q-421ein


https://wolfram.com/xid/0giv1q-stpkgg

Specify scaled offsets from the ordinary coordinates:

https://wolfram.com/xid/0giv1q-u898xy


https://wolfram.com/xid/0giv1q-nn6a0g

Points can be Dynamic:

https://wolfram.com/xid/0giv1q-6gl1zz

Regions (10)
Embedding dimension is the dimension of the space in which the ball lives:

https://wolfram.com/xid/0giv1q-y220

Geometric dimension is the dimension of the shape itself:

https://wolfram.com/xid/0giv1q-bx9tom


https://wolfram.com/xid/0giv1q-6e5svh

https://wolfram.com/xid/0giv1q-3pjv4f

Get conditions for point membership:

https://wolfram.com/xid/0giv1q-2p4iz


https://wolfram.com/xid/0giv1q-se0twe

https://wolfram.com/xid/0giv1q-e06l44


https://wolfram.com/xid/0giv1q-gwq4b4


https://wolfram.com/xid/0giv1q-oknxhk


https://wolfram.com/xid/0giv1q-8idvu6

https://wolfram.com/xid/0giv1q-lkl9xy

The distance to the nearest point for a 2D ball:

https://wolfram.com/xid/0giv1q-da2zys

The equidistance contours for a 3D ball:

https://wolfram.com/xid/0giv1q-ew8anh


https://wolfram.com/xid/0giv1q-bfhwbm

https://wolfram.com/xid/0giv1q-0f4ou3


https://wolfram.com/xid/0giv1q-g7ezsg


https://wolfram.com/xid/0giv1q-d7g53y

https://wolfram.com/xid/0giv1q-mtue

Nearest points to an enclosing sphere:

https://wolfram.com/xid/0giv1q-e29k5d

https://wolfram.com/xid/0giv1q-5ksoo8

https://wolfram.com/xid/0giv1q-uv1cfm


https://wolfram.com/xid/0giv1q-dym4fu

https://wolfram.com/xid/0giv1q-i3tfrr


https://wolfram.com/xid/0giv1q-l3exhn


https://wolfram.com/xid/0giv1q-bx1r15


https://wolfram.com/xid/0giv1q-fivgav

https://wolfram.com/xid/0giv1q-banwkr


https://wolfram.com/xid/0giv1q-nf9ton

https://wolfram.com/xid/0giv1q-hyz4dq

Solve equations in a ball region:

https://wolfram.com/xid/0giv1q-bnrw6

https://wolfram.com/xid/0giv1q-c7lkth

Applications (3)Sample problems that can be solved with this function
Find the minimum surface area for a ball with volume :

https://wolfram.com/xid/0giv1q-opgf2q

https://wolfram.com/xid/0giv1q-bnifa8

Total mass for a ball region with density given by :

https://wolfram.com/xid/0giv1q-h2ora

https://wolfram.com/xid/0giv1q-p48pf6

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

https://wolfram.com/xid/0giv1q-u48x65

https://wolfram.com/xid/0giv1q-6bgb0m


https://wolfram.com/xid/0giv1q-6fqwu0


https://wolfram.com/xid/0giv1q-8she50

Properties & Relations (5)Properties of the function, and connections to other functions
Disk is a special case of Ball:

https://wolfram.com/xid/0giv1q-f599a7


https://wolfram.com/xid/0giv1q-r6dlf

Sphere is the RegionBoundary of Ball:

https://wolfram.com/xid/0giv1q-h6x47m

Ellipsoid is a generalization of Ball:

https://wolfram.com/xid/0giv1q-bgriyc

https://wolfram.com/xid/0giv1q-yms35e

ImplicitRegion can represent any Ball:

https://wolfram.com/xid/0giv1q-hw5ae3

https://wolfram.com/xid/0giv1q-ki4s56

Ball is a norm ball for the Euclidean norm:

https://wolfram.com/xid/0giv1q-mysp6a

https://wolfram.com/xid/0giv1q-b56hh2

Wolfram Research (2014), Ball, Wolfram Language function, https://reference.wolfram.com/language/ref/Ball.html.
Text
Wolfram Research (2014), Ball, Wolfram Language function, https://reference.wolfram.com/language/ref/Ball.html.
Wolfram Research (2014), Ball, Wolfram Language function, https://reference.wolfram.com/language/ref/Ball.html.
CMS
Wolfram Language. 2014. "Ball." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Ball.html.
Wolfram Language. 2014. "Ball." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Ball.html.
APA
Wolfram Language. (2014). Ball. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Ball.html
Wolfram Language. (2014). Ball. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Ball.html
BibTeX
@misc{reference.wolfram_2025_ball, author="Wolfram Research", title="{Ball}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/Ball.html}", note=[Accessed: 10-July-2025
]}
BibLaTeX
@online{reference.wolfram_2025_ball, organization={Wolfram Research}, title={Ball}, year={2014}, url={https://reference.wolfram.com/language/ref/Ball.html}, note=[Accessed: 10-July-2025
]}