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| :''Not to be confused with the [[conical surface]]. For other uses, see [[Cone (disambiguation)]].''
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| {{Refimprove|date=October 2009}}
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| [[File:Cone 3d.png|thumb|250px|right|A right circular cone and an oblique circular cone]]
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| A '''cone''' is a three-[[dimension|dimensional]] [[geometric shape]] that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the [[Apex (geometry)|apex]] or vertex.
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| More precisely, it is the solid figure bounded by a base in a plane and by a surface (called the lateral surface) formed by the [[locus (mathematics)|locus]] of all straight line segments joining the apex to the [[perimeter]] of the base. The term "cone" sometimes refers just to the surface of this solid figure, or just to the lateral surface.
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| The axis of a cone is the straight line (if any), passing through the apex, about which the base has a [[rotational symmetry]].
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| In common usage in elementary [[geometry]], cones are assumed to be '''right circular''', where ''circular'' means that the base is a [[circle]] and ''right'' means that the axis passes through the centre of the base [[perpendicular|at right angles]] to its plane. Contrasted with right cones are oblique cones, in which the axis does not pass perpendicularly through the centre of the base.<ref name="MathWorld">{{MathWorld |urlname=Cone |title=Cone}}</ref> In general, however, the base may be any shape 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).
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| A cone with a [[polygon]]al base is called a [[Pyramid (geometry)|pyramid]].<ref>[http://www.andrews.edu/~calkins/math/webtexts/geom09.htm ''A Review of Basic Geometry'']</ref>
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| == Other mathematical meanings ==
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| [[File:DoubleCone.png|thumb|right|A double cone (not shown infinitely extended)]]
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| In mathematical usage, the word "cone" is used also for an 'infinite cone', the union of a [[set (mathematics)|set]] of [[half-line]]s that start at a common apex point and go through a base. An infinite cone is not bounded by its base but extends to infinity. A 'doubly infinite cone', or 'double cone', is the union of a set of [[straight line]]s that pass through a common apex point and go through a base, therefore double infinite cones extend symmetrically on both sides of the apex.
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| The boundary of an infinite or doubly infinite cone is a [[conical surface]], and the intersection of a plane with this surface is a [[conic section]]. For infinite cones, the word ''axis'' again usually refers to the axis of rotational symmetry (if any). Either half of a double cone on one side of the apex is called a 'nappe'.
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| Depending on the context, "cone" may also mean specifically a [[convex cone]] or a [[projective cone]].
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| == Further terminology ==
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| The perimeter of the base of a cone is called the 'directrix', and each of the line segments between the directrix and apex is a 'generatrix' of the lateral surface. (For the connection between this sense of the term "directrix" and the [[Directrix (conic section)|directrix]] of a conic section, see [[Dandelin spheres]].)
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| The volume and the surface area for a straight cone are described in the [[#Geometry|geometry]] section below.
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| The 'base radius' of a circular cone is the [[radius]] of its base; often this is simply called the radius of the cone. The [[aperture]] of a right circular cone is the maximum angle between two generatrix lines; if the generatrix makes an angle ''θ'' to the axis, the aperture is 2''θ''.
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| A cone with its apex cut off by a plane is called a "truncated cone"; if the truncation plane is parallel to the cone's base, it is called a [[frustum]]. An 'elliptical cone' is a cone with an [[ellipse|elliptical]] base. A 'generalized cone' is the surface created by the set of lines passing through a vertex and every point on a boundary (also see [[visual hull]]).
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| == Geometry ==
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| <!-- The formulae are correct. Please check your work before editing. --><!-- Please put proofs and derivations in [[cone (geometry) proofs]] -->
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| ===Surface area===
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| The [[lateral surface]] area of a right circular cone is <math>LSA = \pi r l</math> where <math>r</math> is the radius of the circle at the bottom of the cone and <math>l</math> is the lateral height of the cone (given by the [[Pythagorean theorem]] <math>l=\sqrt{r^2+h^2}</math> where <math>h</math> is the height of the cone). The surface area of the bottom circle of a cone is the same as for any circle, <math>\pi r^2</math>. Thus the total surface area of a right circular cone is:
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| :<math>SA=\pi r^2+\pi r l</math> or
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| :<math>SA=\pi r(r+l)</math>
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| === Volume ===
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| [[File:A cone being held by a woman with volume formula.jpg|thumb|Volume is calculated by multiplying the area of the base circle times the height, and multiplying by one third.]]
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| The [[volume]] <math>V</math> of any conic solid is one third of the product of the area of the base <math>B</math> and the height <math>H</math> (the perpendicular distance from the base to the apex).
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| :<math>V = \frac{1}{3} B H</math>
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| In modern mathematics, this formula can easily be computed using calculus – it is, up to scaling, the integral <math>\int x^2 dx = \tfrac{1}{3} x^3.</math> Without using calculus, the formula can be proven by comparing the cone to a pyramid and applying [[Cavalieri's principle]] – specifically, comparing the cone to a (vertically scaled) right square pyramid, which forms one third of a cube. This formula cannot be proven without using such infinitesimal arguments – unlike the 2-dimensional formulae for polyhedral area, though similar to the area of the circle – and hence admitted less rigorous proofs before the advent of calculus, with the ancient Greeks using the [[method of exhaustion]]. This is essentially the content of [[Hilbert's third problem]] – more precisely, not all polyhedral pyramids are ''scissors congruent'' (can be cut apart into finite pieces and rearranged into the other), and thus volume cannot be computed purely by using a decomposition argument.
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| === Center of mass ===
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| The [[center of mass]] of a conic solid of uniform density lies one-quarter of the way from the center of the base to the vertex, on the straight line joining the two.
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| === Right circular cone ===
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| For a circular cone with radius ''R'' and height ''H'', the formula for volume becomes
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| :<math> V = \int_0^H r^2 \pi \, dh </math> | |
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| where ''r'' is the radius of the cone at height ''h'' measured from the apex:
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| :<math> r= R \frac{h}{H} </math>
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| Thus:
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| :<math> V = \int_0^H \left[R \frac{h}{H}\right]^2 \pi \, dh </math>
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| Thus:
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| :<math>V = \frac{1}{3} \pi R^2 H. </math>
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| For a right circular cone, the surface [[area]] <math>A</math> is
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| :<math>A =\pi R^2 + \pi R S\,</math> where <math>S = \sqrt{R^2 + H^2}</math> is the [[slant height]].
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| The first term in the area formula, <math>\pi R^2</math>, is the area of the base, while the second term, <math>\pi R S</math>, is the area of the lateral surface.
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| A right circular cone with height <math>h</math> and aperture <math>2\theta</math>, whose axis is the <math>z</math> coordinate axis and whose apex is the origin, is described parametrically as
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| :<math>F(s,t,u) = \left(u \tan s \cos t, u \tan s \sin t, u \right)</math>
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| where <math>s,t,u</math> range over <math>[0,\theta)</math>, <math>[0,2\pi)</math>, and <math>[0,h]</math>, respectively.
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| In [[Implicit function|implicit]] form, the same solid is defined by the inequalities
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| :<math>\{ F(x,y,z) \leq 0, z\geq 0, z\leq h\},</math>
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| where
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| :<math>F(x,y,z) = (x^2 + y^2)(\cos\theta)^2 - z^2 (\sin \theta)^2.\,</math>
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| More generally, a right circular cone with vertex at the origin, axis parallel to the vector <math>d</math>, and aperture <math>2\theta</math>, is given by the implicit [[vector calculus|vector]] equation <math>F(u) = 0</math> where
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| :<math>F(u) = (u \cdot d)^2 - (d \cdot d) (u \cdot u) (\cos \theta)^2</math> or <math>F(u) = u \cdot d - |d| |u| \cos \theta</math> | |
| where <math>u=(x,y,z)</math>, and <math>u \cdot d</math> denotes the [[dot product]].
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| == Projective geometry ==
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| [[File:Australia Square building in George Street Sydney.jpg|thumb|In [[projective geometry]], a [[Cylinder (geometry)|cylinder]] is simply a cone whose apex is at infinity, which corresponds visually to a cylinder in perspective appearing to be a cone towards the sky.]]
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| In [[projective geometry]], a cylinder is simply a cone whose apex is at infinity. Intuitively, if one keeps the base fixed and takes the limit as the apex goes to infinity, one obtains a cylinder, the angle of the side increasing as [[arctan]], in the limit forming a [[right angle]].
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| This is useful in the definition of [[degenerate conic]]s, which require considering the [[cylindrical conic]]s.
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| == See also ==
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| * [[Conic section]]
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| * [[Cone (linear algebra)]]
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| * [[Cone (topology)]]
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| * [[Bicone]]
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| * [[Democritus]]
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| * [[Hyperboloid]]
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| * [[Quadric]]
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| * [[Ruled surface]]
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| * [[Pyrometric cone]]
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| * [[Cylinder]]
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| == References ==
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| {{Reflist}}
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| == External links ==
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| {{Commons category|Cone (geometry)}}
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| * {{MathWorld |urlname=GeneralizedCone |title=Generalized Cone}}
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| * [http://www.mathsisfun.com/geometry/cone.html Spinning Cone] from [[Math Is Fun]]
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| * [http://www.korthalsaltes.com/model.php?name_en=cone Paper model cone]
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| * [http://mathforum.org/library/drmath/view/55017.html Lateral surface area of an oblique cone]
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| * [http://www.cut-the-knot.org/Curriculum/Geometry/ConicSections.shtml Cut a Cone] An interactive demonstration of the intersection of a cone with a plane
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| [[Category:Elementary shapes]]
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| [[Category:Surfaces]]
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