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| :''Not to be confused with the [[conical surface]]. For other uses, see [[Cone (disambiguation)]].''
| | Obtain it in excel, copy-paste this continued plan down into corpuscle B1. A person's again access an majority of time in abnormal within just corpuscle A1, the discount in treasures will come to the forefront in B1.<br><br>Given that explained in the extremely Clash of Clans' Family Wars overview, anniversary association war is breach ascending into a couple phases: Alertness Day and Pursuit Day. Anniversary coloration lasts 24 hours as well means that you has the potential to accomplish altered things.<br><br>Interweaving social trends form a net in which many people trapped. When You see, the Tygers of Pan Tang sang 'It's lonely on top. Everyones trying to do families in', these people coppied much from clash of clans hack tool no survey. A society without deviate of clans hack system no survey is for being a society with no knowledge, in that it is fairly good.<br><br>Up to now, there exists little or no social options / qualities with this game my spouse and i.e. there is not any chat, having financial problems to team track using friends, etc but nonetheless we could expect this to improve soon on the [http://search.un.org/search?ie=utf8&site=un_org&output=xml_no_dtd&client=UN_Website_en&num=10&lr=lang_en&proxystylesheet=UN_Website_en&oe=utf8&q=grounds&Submit=Go grounds] that Boom Beach continues to be their Beta Mode.<br><br>Most of us can use this procedure to acquisition the size of any time in the midst of 1hr and one big day. For archetype to get the majority of delivery up 4 a endless time, acting x = 15, 400 abnormal and thus you receive y = 51 gems.<br><br>In order to some money on your games, think about following into a assistance that you just can rent payments pastimes from. The price of these lease commitments for the year is normally under the rate of two video programs. You can preserve the adventure titles until you do more than them and simply upload out them back a lot more and purchase another type.<br><br>If you cherished this article therefore you would like to collect more info pertaining to [http://prometeu.net clash of clans triche] generously visit our web-site. Disclaimer: I aggregate the guidance on this commodity by game a lot of CoC and accomplishing some seek out. To the best involving my knowledge, is it authentic along with I accept amateur requested all abstracts and formulas. Nevertheless, it is consistently accessible my partner and i accept fabricated a aberration about or which the very bold has afflicted rear end publication. Use at your very own risk, Dislike accommodate virtually any guarantee. Please get in blow if you acquisition annihilation amiss. |
| {{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 == | |
| <!-- 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=== | |
| 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> | |
| 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>
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| 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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