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Cone [{{x1,y1,z1},{x2,y2,z2}},r]

represents a cone with a base of radius r centered at (x1,y1,z1) and a tip at (x2,y2,z2).

Cone [{{x1,y1,z1},{x2,y2,z2}}]

represents a cone with a base of radius 1.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Graphics  
Specification  
Styling  
Coordinates  
Regions  
Applications  
Properties & Relations  
Neat Examples  
See Also
Related Guides
Related Links
History
Cite this Page

Cone [{{x1,y1,z1},{x2,y2,z2}},r]

represents a cone with a base of radius r centered at (x1,y1,z1) and a tip at (x2,y2,z2).

Cone [{{x1,y1,z1},{x2,y2,z2}}]

represents a cone with a base of radius 1.

Details and Options

  • Cone can be used as a geometric region and a graphics primitive.
  • Cone [] is equivalent to Cone [{{0,0,-1},{0,0,1}}].
  • Cone represents a filled cone region where and the vectors are orthogonal with TemplateBox[{{v, _, 1}}, Norm]=TemplateBox[{{v, _, 2}}, Norm]=1, and and .
  • Cone can be used in Graphics3D .
  • In graphics, the points pi and radii r can be Scaled and Dynamic expressions.
  • Graphics rendering is affected by directives such as EdgeForm , FaceForm , Specularity , Opacity , and color.
  • Cone [{spec1,spec2,…},{r1,r2,…}] represents a collection of cones with specifications speci and base radii ri.

Examples

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Basic Examples  (4)

A unit radius and two units height cone:

Wolfram Language code: Graphics3D[Cone[]]

A cone from the origin to {1,1,1} with radius 1/2 at its base:

Wolfram Language code: Graphics3D[Cone[{{0, 0, 0}, {1, 1, 1}}, 1 / 2]]

Differently styled cones:

Wolfram Language code: {Graphics3D[{Yellow, Cone[]}], Graphics3D[{EdgeForm[Thick], Cone[]}], Graphics3D[{EdgeForm[Dashed], Cone[]}], Graphics3D[{EdgeForm[Directive[Thick, Dashed, Blue]], Orange, Cone[]}]}

Volume and centroid:

Wolfram Language code: Volume[Cone[{{Subscript[x, 1], Subscript[y, 1], Subscript[z, 1]}, {Subscript[x, 2], Subscript[y, 2], Subscript[z, 2]}}, r]]
Wolfram Language code: RegionCentroid[Cone[{{Subscript[x, 1], Subscript[y, 1], Subscript[z, 1]}, {Subscript[x, 2], Subscript[y, 2], Subscript[z, 2]}}, r]]

Scope  (22)

Graphics  (12)

Specification  (5)

If no radius is specified, it is assumed to be 1:

Wolfram Language code: Graphics3D[Cone[{{0, 0, 0}, {0, 0, 1}}]]

Cone with different radii:

Wolfram Language code: Graphics3D[{Cone[{{0, 0, 3}, {0, 0, -3}}, 1], Cone[{{5, 0, 3}, {5, 0, -3}}, 3]}]

Cone with different directions:

Wolfram Language code: Graphics3D[{Cone[{{0, 0, 0}, {1, 0, 0}}], Cone[{{1, 1, 1}, {2, 3, 1}}]}]

Short form for a cone centered at the origin with a base radius 1:

Wolfram Language code: Graphics3D[Cone[], Axes -> True]

Multiple cones:

Wolfram Language code: s = {{0, 0, 0}, {0, 0, 2}};
Wolfram Language code: Graphics3D[Cone[{s, s + 2}]]

Styling  (5)

Color directives specify the face colors of cones:

Wolfram Language code: Table[Graphics3D[{c, Cone[]}], {c, {Red, Green, Blue, Yellow}}]

FaceForm and EdgeForm can be used to specify the styles of the faces and edges:

Wolfram Language code: Graphics3D[{FaceForm[Pink], EdgeForm[Directive[Dashed, Thick, Blue]], Cone[]}]

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

Wolfram Language code: Graphics3D[{FaceForm[Yellow, Blue], Cone[]}, PlotRange -> {{-1, 1}, {-.2, 1}, {-1, 1}}]

Cones with different specular exponents:

Wolfram Language code: Table[Graphics3D[{Orange, Specularity[White, n], Cone[]}, Lighting -> {{"Point", White, Scaled[{2, -1, 1.2}]}}], {n, {5, 20, 100}}]

White cone that glows red:

Wolfram Language code: Graphics3D[{Glow[Red], Black, Cone[]}]

Opacity specifies the face opacity:

Wolfram Language code: Table[Graphics3D[{Opacity[o], Cone[]}], {o, {0.3, 0.5, 0.9}}]

Coordinates  (2)

Use Scaled coordinates:

Wolfram Language code: Graphics3D[Cone[{Scaled[{.5, .5, .2}], Scaled[{.5, .5, .8}]}, 3], PlotRange -> {{0, 10}, {0, 10}, {0, 10}}, Axes -> True]

Use Scaled radius:

Wolfram Language code: Graphics3D[Cone[{{0, 0, -1}, {0, 0, 1}}, Scaled[0.25]], Axes -> True, PlotRange -> 1]

Regions  (10)

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

Wolfram Language code: ℛ = Cone[{{0, 0, 0}, {1, 1, 1}}, 2];
Wolfram Language code: RegionEmbeddingDimension[ℛ]

Geometric dimension is the dimension of the shape itself:

Wolfram Language code: RegionDimension[ℛ]

Membership testing:

Wolfram Language code: ℛ = Cone[{{0, 0, 0}, {1, 1, 1}}, 2];
Wolfram Language code: {RegionMember[ℛ, {(1/3), (1/3), (1/3)}], RegionMember[ℛ, {5, 5, 5}]}

Get conditions for membership:

Wolfram Language code: RegionMember[ℛ, {x, y, z}]

Volume:

Wolfram Language code: ℛ = Cone[{{0, 0, 0}, {1, 1, 1}}, 2];
Wolfram Language code: {Volume[ℛ], RegionMeasure[ℛ]}

Centroid:

Wolfram Language code: c = RegionCentroid[ℛ]
Wolfram Language code: Graphics3D[{{Opacity[0.5], LightBlue, ℛ}, {PointSize[Large], Red, Point[c]}}]

Distance from a point:

Wolfram Language code: ℛ = Cone[{{0, 0, 0}, {1, 1, 1}}, 2];
Wolfram Language code: {RegionDistance[ℛ, {1, 2, 3}], RegionDistance[ℛ, {(1/3), (1/4), (1/5)}]}

The equidistance contours for a cone:

Wolfram Language code: ContourPlot3D[Evaluate@RegionDistance[ℛ, {x, y, z}], {x, -3, 3}, {y, -3, 2.5}, {z, -2.5, 3}, Mesh -> None, Contours -> {0.25, 0.5, 1}, ContourStyle -> ColorData[94, "ColorList"], Lighting -> "Neutral", BaseStyle -> Opacity[0.5], BoxRatios -> Automatic]

Signed distance from a point:

Wolfram Language code: ℛ = Cone[{{0, 0, 0}, {1, 1, 1}}, 2];
Wolfram Language code: {SignedRegionDistance[ℛ, {1, 2, 3}], SignedRegionDistance[ℛ, {(1/3), (1/4), (1/5)}]}

Nearest point in the region:

Wolfram Language code: ℛ = Cone[{{0, 0, 0}, {1, 1, 1}}, 2];
Wolfram Language code: {RegionNearest[ℛ, {1, 2, 3}], RegionNearest[ℛ, {(1/3), (1/4), (1/5)}]}

Nearest points to an enclosing sphere:

Wolfram Language code: spherePoints[{n_, m_}, c_, r_] := Flatten[Table[c + r{Cos[k 2π / n]Sin[l π / m], Sin[k 2π / n]Sin[l π / m], Cos[l π / m]}, {k, 0., n - 1}, {l, 0., m - 1}], 1];
Wolfram Language code: pl = spherePoints[{16, 8}, RegionCentroid[ℛ], 3]; npl = Table[RegionNearest[ℛ, p], {p, pl}];
Wolfram Language code: Legended[Graphics3D[{ℛ, {Thin, Gray, Line[Transpose[{pl, npl}]]}, {Red, Point[pl]}, {PointSize[Medium], Blue, Point[npl]}}, Lighting -> "Neutral", Boxed -> False], PointLegend[{Red, Blue}, {"start", "nearest"}]]

A cone is bounded:

Wolfram Language code: ℛ = Cone[{{0, 0, 0}, {1, 1, 1}}, 1];
Wolfram Language code: BoundedRegionQ[ℛ]

Find its range:

Wolfram Language code: r = RegionBounds[ℛ]
Wolfram Language code: Graphics3D[{{EdgeForm[White], Opacity[0.2, Yellow], Cuboid@@Transpose[r]}, ℛ}, Boxed -> False]

Integrate over a cone region:

Wolfram Language code: ℛ = Cone[{{0, 0, 0}, {1, 1, 1}}, 2];
Wolfram Language code: Integrate[x + y + z, {x, y, z}∈ℛ]

Optimize over a cone region:

Wolfram Language code: ℛ = Cone[{{0, 0, 0}, {1, 1, 1}}, 2];
Wolfram Language code: MinValue[{x y z - x y, {x, y, z}∈ℛ}, {x, y, z}]

Solve equations in a cone region:

Wolfram Language code: ℛ = Cone[{{0, 0, 0}, {1, 1, 1}}, 2];
Wolfram Language code: Reduce[x^2 + y^2 + z^2 == 1 && x - y - z == -1 && z^2 == x y && {x, y, z}∈ℛ, {x, y, z}]

Applications  (5)

Find the minimum surface area for a cone with volume :

Wolfram Language code: ℛ = Cone[{{0, 0, 0}, {0, 0, h}}, r];
Wolfram Language code: Minimize[{Area[RegionBoundary[ℛ]], Volume[ℛ] == π / 3 && r > 0 && h > 0}, {r, h}, Reals]

Compare with some other cones of the same volume:

Wolfram Language code: Graphics3D[#, PlotLabel -> Area[RegionBoundary[#]]]& /@ Diagonal@Table[ℛ, {r, {1 / (4Sqrt[2]), 1 / Sqrt[2], 2 / Sqrt[2]}}, {h, {8, 2, 1 / 2}}]

Define a region by the intersection of a cone and a plane:

Wolfram Language code: Subscript[ℛ, 1] = Cone[{{0, 0, 0}, {0, 0, 2}}, 2]; Subscript[ℛ, 2] = InfinitePlane[{0, 0, 0}, {{1, 1, 1}, {0, 1, 1}}];
Wolfram Language code: i = RegionIntersection[Subscript[ℛ, 1], Subscript[ℛ, 2]]//DiscretizeRegion

Visualize the intersection:

Wolfram Language code: g = Graphics3D[{{Opacity[0.3], Darker@Green, Subscript[ℛ, 1]}, {Opacity[0.5], LightRed, Subscript[ℛ, 2]}}, Boxed -> False, Lighting -> "Neutral"];
Wolfram Language code: Show[g, i]

A simple 3D arrow chart:

Wolfram Language code: data = RandomReal[{1, 10}, {12, 4}];
Wolfram Language code: Graphics3D[MapIndexed[{Hue[(Last[#2] - 1) / 4], Cylinder[{Append[{1, 2}#2, 0], Append[{1, 2}#2, #1]}, .4], Cone[{Append[{1, 2}#2, #1], Append[{1, 2}#2, #1 + 2]}, .6]}&, data, {2}], Axes -> {False, False, True}, Lighting -> "Neutral"]

Define a ChartElementFunction based on Cone :

Wolfram Language code: ConeBar[{{xmin_, xmax_}, {ymin_, ymax_}, {zmin_, zmax_}}, ___] := Cone[{{(xmin + xmax) / 2, (ymin + ymax) / 2, zmin}, {(xmin + xmax) / 2, (ymin + ymax) / 2, zmax}}, Min[(xmax - xmin) / 2, (ymax - ymin) / 2]]
Wolfram Language code: data = RandomReal[1, {2, 5}];
Wolfram Language code: BarChart3D[data, ChartElementFunction -> ConeBar]

BarChart3D uses Cone to produce 3D bar charts:

Wolfram Language code: BarChart3D[data, ChartElementFunction -> "Cone"]

Histogram3D can similarly use Cone :

Wolfram Language code: Histogram3D[RandomReal[NormalDistribution[0, 1], {500, 2}], 10, ChartElementFunction -> "Cone"]

Use Cone to display bubbles in BubbleChart3D :

Wolfram Language code: BubbleChart3D[RandomReal[1, {3, 10, 4}], ChartElements -> Graphics3D[Cone[]]]

Properties & Relations  (6)

Use Scale to get an elliptical cone:

Wolfram Language code: Graphics3D[Scale[Cone[], {2, 4, 3}, {0, 0, 0}], Axes -> True]

Cone is used as a 3D arrowhead in Arrow :

Wolfram Language code: Graphics3D[{Arrowheads[.4], Blue, Arrow[Tube[{{0, 0, 0}, {2, 1, 1}}, 0.1, VertexColors -> {Red, Blue}]]}]

Cone is a special case of Tube :

Wolfram Language code: {Graphics3D[Cone[{{0, 0, 0}, {1, 1, 1}}, 1]], Graphics3D[{CapForm["Butt"], Tube[{{0, 0, 0}, {1, 1, 1}}, {1, 0}]}]}

Get a truncated cone by specifying different radii in Tube :

Wolfram Language code: Graphics3D[{CapForm["Butt"], Tube[{{0, 0, 0}, {1, 1, 1}}, {1, 1 / 2}]}]

A parametric specification of a cone shell generated using ParametricPlot3D :

Wolfram Language code: ParametricPlot3D[{Cos[θ] z, Sin[θ]z, z}, {θ, 0, 2 π}, {z, -1, 0}, Mesh -> None]

An implicit specification of a cone shell generated by ContourPlot3D :

Wolfram Language code: ContourPlot3D[x ^ 2 + y ^ 2 == z ^ 2, {x, -1, 1}, {y, -1, 1}, {z, -1, 0}, Mesh -> None]

ImplicitRegion can represent any Cone region:

Wolfram Language code: Subscript[ℛ, 1] = ImplicitRegion[0 ≤ Subscript[t, 1] + Subscript[t, 2] + Subscript[t, 3] ≤ 3 && Subsuperscript[t, 1, 2] + Subsuperscript[t, 2, 2] + Subscript[t, 2] (12 - 7 Subscript[t, 3]) + Subscript[t, 1] (12 - 7 Subscript[t, 2] - 7 Subscript[t, 3]) + Subscript[t, 3] (12 + Subscript[t, 3]) ≤ 18, {Subscript[t, 1], Subscript[t, 2], Subscript[t, 3]}]; Subscript[ℛ, 2] = Cone[{{0, 0, 0}, {1, 1, 1}}, 2];
Wolfram Language code: RegionEqual[Subscript[ℛ, 1], Subscript[ℛ, 2]]

Neat Examples  (3)

Random unit cones:

Wolfram Language code: Graphics3D[Table[{EdgeForm[Opacity[.3]], Hue[RandomReal[]], Cone[RandomReal[10, {2, 3}]]}, {20}]]

Sweep a cone around an axis:

Wolfram Language code: Graphics3D[{Opacity[0.3], EdgeForm[], Table[{ColorData["Rainbow"][Rescale[c, {0, 2Pi}]], GeometricTransformation[Cone[], RotationTransform[c, {-1, 2, -6}, {1, 0, 0}]]}, {c, 0, 2Pi, 2Pi / 18}]}]

Nested transparent cones:

Wolfram Language code: Graphics3D[{Opacity[.3], EdgeForm[Opacity[.3]], Table[Cone[{{0, 0, 0}, {0, 0, 2r}}, r], {r, 1, 5}]}, Boxed -> False]
Wolfram Research (2008), Cone, Wolfram Language function, https://reference.wolfram.com/language/ref/Cone.html (updated 2014).

Text

Wolfram Research (2008), Cone, Wolfram Language function, https://reference.wolfram.com/language/ref/Cone.html (updated 2014).

CMS

Wolfram Language. 2008. "Cone." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/Cone.html.

APA

Wolfram Language. (2008). Cone. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Cone.html

BibTeX

@misc{reference.wolfram_2026_cone, author="Wolfram Research", title="{Cone}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/Cone.html}", note=[Accessed: 13-August-2026]}

BibLaTeX

@online{reference.wolfram_2026_cone, organization={Wolfram Research}, title={Cone}, year={2014}, url={https://reference.wolfram.com/language/ref/Cone.html}, note=[Accessed: 13-August-2026]}
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