Editing Cone (section)

{{short description|Geometric shape}}
{{other uses|Cone (disambiguation)}}
{{distinguish|Conical surface|Truncated dome}}
{{Infobox polyhedron
| name          = Cone
| image         = Cone with labeled Radius, Height, Angle and Side.svg|
| caption       = A right circular cone with the radius of its base ''r'', its height ''h'', its slant height ''c'' and its angle ''θ''.
| type          = Solid figure
| faces         = 1 circular face and 1 conic surface
| euler         = 2
| symmetry      = [[Orthogonal group|{{math|O(2)}}]]
| surface_area  = {{math|[[Pi|{{pi}}]]''r''<sup>2</sup> + [[Pi|{{pi}}]]''rℓ''}}
| volume        = {{math|([[Pi|{{pi}}]]''r''<sup>2</sup>''h'')/3}}
}}
[[File:Cone 3d.png|thumb|upright=1.2|A right circular cone and an oblique circular cone]]
[[File:DoubleCone.png|thumb|A double cone, not infinitely extended]]

In [[geometry]], a '''cone''' is a [[three-dimensional figure]] that tapers smoothly from a [[plane (geometry)|flat]] base (typically a [[circle]]) to a point not contained in the base, called the ''[[Apex (geometry)|apex]]'' or ''[[vertex (geometry)|vertex]]''.

A cone is formed by a set of [[line segment]]s, [[Ray (geometry)|half-line]]s, or [[Line (geometry)|line]]s connecting a common point, the apex, to all of the points on a base.  In the case of line segments, the cone does not extend beyond the base, while in the case of half-lines, it extends infinitely far. In the case of lines, the cone extends infinitely far in both directions from the apex, in which case it is sometimes called a ''double cone''{{anchor|Double}}. Each of the two halves of a double cone split at the apex is called a ''nappe''{{anchor|Nappe}}.

Depending on the author, the base may be restricted to a circle, any one-dimensional [[quadratic form]] in the plane, any closed [[one-dimensional space|one-dimensional figure]], or any of the above plus all the enclosed points. If the enclosed points are included in the base, the cone is a [[solid geometry|solid object]]; otherwise it is an [[open surface]], a [[two-dimensional]] object in three-dimensional space. In the case of a solid object, the boundary formed by these lines or partial lines is called the ''lateral surface''; if the lateral surface is [[Unbounded set|unbounded]], it is a ''[[conical surface]]''.

The [[rotational symmetry|axis]] of a cone is the straight line passing through the apex about which the cone has a [[circular symmetry]]. {{anchor|Right circular}}In common usage in elementary geometry, cones are assumed to be ''right circular'', i.e., with a circle base [[perpendicular]] to the axis.<ref name=":1">{{Cite book|url=https://books.google.com/books?id=UyIfgBIwLMQC|title=The Mathematics Dictionary|last1=James|first1=R. C. |author-link1=Robert C. James |last2=James|first2=Glenn|date=1992-07-31|publisher=Springer Science & Business Media|isbn=9780412990410|pages=74–75}}</ref> If the cone is right circular the intersection of a plane with the lateral surface is a [[conic section]]. In general, however, the base may be any shape<ref name="grunbaum">Grünbaum, ''[[Convex Polytopes]]'', second edition, p. 23.</ref> and the apex may lie anywhere (though it is usually assumed that the base is bounded and therefore has finite [[area (geometry)|area]], and that the apex lies outside the plane of the base). Contrasted with right cones are ''oblique cones'', in which the axis passes through the centre of the base non-perpendicularly.<ref name="MathWorld">{{MathWorld |urlname=Cone |title=Cone}}</ref> 

Depending on context, ''cone'' may refer more narrowly to either a [[convex cone]] or [[projective cone]].
Cones can be generalized to [[Dimension#Additional dimensions|higher dimensions]].