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cone wikipedia 403 captures 04 jan 2004 29 sep 2026 aug sep oct 30 2021 2022 2023 success fail about this capture timestamps the wayback machine http web archive org web 20220930125458 https en wikipedia org wiki cone cone from wikipedia the free encyclopedia jump to navigation jump to search geometric shape for other uses see cone disambiguation not to be confused with conical surface a right circular cone and an oblique circular cone a double cone not shown infinitely extended 3d model of a cone a cone is a three dimensional geometric shape that tapers smoothly from a flat base frequently though not necessarily circular to a point called the apex or vertex a cone is formed by a set of line segments half lines or lines connecting a common point the apex to all of the points on a base that is in a plane that does not contain the apex depending on the author the base may be restricted to be a circle any one dimensional quadratic form in the plane any closed 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 object otherwise it is 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 it is a conical surface 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 either half of a double cone on one side of the apex is called a nappe the axis of a cone is the straight line if any passing through the apex about which the base and the whole cone has a circular symmetry 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 at right angles to its plane 1 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 2 and the apex may lie anywhere though it is usually assumed that the base is bounded and therefore has finite 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 3 a cone with a polygonal base is called a pyramid depending on the context cone may also mean specifically a convex cone or a projective cone cones can also be generalized to higher dimensions contents 1 further terminology 2 measurements and equations 2 1 volume 2 2 center of mass 2 3 right circular cone 2 3 1 volume 2 3 2 slant height 2 3 3 surface area 2 3 4 circular sector 2 3 5 equation form 2 4 elliptic cone 3 projective geometry 4 higher dimensions 5 see also 6 notes 7 references 8 external links further terminology edit 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 or generating line of the lateral surface for the connection between this sense of the term directrix and the directrix of a conic section see dandelin spheres 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 illustration from problemata mathematica published in acta eruditorum 1734 a cone with a region including 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 1 an elliptical cone is a cone with an elliptical base 1 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 measurements and equations edit volume edit the volume v displaystyle v of any conic solid is one third of the product of the area of the base a b displaystyle a_ b and the height h displaystyle h 4 v 1 3 a b h displaystyle v frac 1 3 a_ b h in modern mathematics this formula can easily be computed using calculus it is up to scaling the integral x 2 d x 1 3 x 3 displaystyle int x 2 dx tfrac 1 3 x 3 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 5 center of mass edit 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 right circular cone edit volume edit for a circular cone with radius r and height h the base is a circle of area π r 2 displaystyle pi r 2 and so the formula for volume becomes 6 v 1 3 π r 2 h displaystyle v frac 1 3 pi r 2 h slant height edit the slant height of a right circular cone is the distance from any point on the circle of its base to the apex via a line segment along the surface of the cone it is given by r 2 h 2 displaystyle sqrt r 2 h 2 where r displaystyle r is the radius of the base and h displaystyle h is the height this can be proved by the pythagorean theorem surface area edit the lateral surface area of a right circular cone is l s a π r l displaystyle lsa pi rl where r displaystyle r is the radius of the circle at the bottom of the cone and l displaystyle l is the slant height of the cone 4 the surface area of the bottom circle of a cone is the same as for any circle π r 2 displaystyle pi r 2 thus the total surface area of a right circular cone can be expressed as each of the following radius and height π r 2 π r r 2 h 2 displaystyle pi r 2 pi r sqrt r 2 h 2 the area of the base plus the area of the lateral surface the term r 2 h 2 displaystyle sqrt r 2 h 2 is the slant height π r r r 2 h 2 displaystyle pi r left r sqrt r 2 h 2 right where r displaystyle r is the radius and h displaystyle h is the height radius and slant height π r 2 π r l displaystyle pi r 2 pi rl π r r l displaystyle pi r r l where r displaystyle r is the radius and l displaystyle l is the slant height circumference and slant height c 2 4 π c l 2 displaystyle frac c 2 4 pi frac cl 2 c 2 c 2 π l displaystyle left frac c 2 right left frac c 2 pi l right where c displaystyle c is the circumference and l displaystyle l is the slant height apex angle and height π h 2 tan θ 2 tan θ 2 sec θ 2 displaystyle pi h 2 tan frac theta 2 left tan frac theta 2 sec frac theta 2 right where θ displaystyle theta is the apex angle and h displaystyle h is the height circular sector edit the circular sector obtained by unfolding the surface of one nappe of the cone has radius r r r 2 h 2 displaystyle r sqrt r 2 h 2 arc length l l c 2 π r displaystyle l c 2 pi r central angle φ in radians ϕ l r 2 π r r 2 h 2 displaystyle phi frac l r frac 2 pi r sqrt r 2 h 2 equation form edit the surface of a cone can be parameterized as f θ h h cos θ h sin θ h displaystyle f theta h h cos theta h sin theta h where θ 0 2 π displaystyle theta in 0 2 pi is the angle around the cone and h r displaystyle h in mathbb r is the height along the cone a right solid circular cone with height h displaystyle h and aperture 2 θ displaystyle 2 theta whose axis is the z displaystyle z coordinate axis and whose apex is the origin is described parametrically as f s t u u tan s cos t u tan s sin t u displaystyle f s t u left u tan s cos t u tan s sin t u right where s t u displaystyle s t u range over 0 θ displaystyle 0 theta 0 2 π displaystyle 0 2 pi and 0 h displaystyle 0 h respectively in implicit form the same solid is defined by the inequalities f x y z 0 z 0 z h displaystyle f x y z leq 0 z geq 0 z leq h where f x y z x 2 y 2 cos θ 2 z 2 sin θ 2 displaystyle f x y z x 2 y 2 cos theta 2 z 2 sin theta 2 more generally a right circular cone with vertex at the origin axis parallel to the vector d displaystyle d and aperture 2 θ displaystyle 2 theta is given by the implicit vector equation f u 0 displaystyle f u 0 where f u u d 2 d d u u cos θ 2 displaystyle f u u cdot d 2 d cdot d u cdot u cos theta 2 or f u u d d u cos θ displaystyle f u u cdot d d u cos theta where u x y z displaystyle u x y z and u d displaystyle u cdot d denotes the dot product elliptic cone edit an elliptical cone quadric surface in the cartesian coordinate system an elliptic cone is the locus of an equation of the form 7 x 2 a 2 y 2 b 2 z 2 displaystyle frac x 2 a 2 frac y 2 b 2 z 2 it is an affine image of the right circular unit cone with equation x 2 y 2 z 2 displaystyle x 2 y 2 z 2 from the fact that the affine image of a conic section is a conic section of the same type ellipse parabola one gets any plane section of an elliptic cone is a conic section obviously any right circular cone contains circles this is also true but less obvious in the general case see circular section the intersection of an elliptic cone with a concentric sphere is a spherical conic projective geometry edit in projective geometry a 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 in projective geometry a cylinder is simply a cone whose apex is at infinity 8 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 this is useful in the definition of degenerate conics which require considering the cylindrical conics according to g b halsted a cone is generated similarly to a steiner conic only with a projectivity and axial pencils not in perspective rather than the projective ranges used for the steiner conic if two copunctual non costraight axial pencils are projective but not perspective the meets of correlated planes form a conic surface of the second order or cone 9 higher dimensions edit the definition of a cone may be extended to higher dimensions see convex cones in this case one says that a convex set c in the real vector space r n is a cone with apex at the origin if for every vector x in c and every nonnegative real number a the vector ax is in c 2 in this context the analogues of circular cones are not usually special in fact one is often interested in polyhedral cones see also edit bicone cone linear algebra cone topology cylinder geometry democritus generalized conic hyperboloid list of shapes pyrometric cone quadric rotation of axes ruled surface translation of axes notes edit a b c james r c james glenn 1992 07 31 the mathematics dictionary springer science business media pp 74 75 isbn 9780412990410 a b grünbaum convex polytopes second edition p 23 weisstein eric w cone mathworld a b alexander daniel c koeberlein geralyn m 2014 01 01 elementary geometry for college students cengage learning isbn 9781285965901 hartshorne robin 2013 11 11 geometry euclid and beyond springer science business media chapter 27 isbn 9780387226767 blank brian e krantz steven george 2006 01 01 calculus single variable springer science business media chapter 8 isbn 9781931914598 protter morrey 1970 p 583 harvtxt error no target citerefprottermorrey1970 help dowling linnaeus wayland 1917 01 01 projective geometry mcgraw hill book company incorporated g b halsted 1906 synthetic projective geometry page 20 references edit protter murray h morrey jr charles b 1970 college calculus with analytic geometry 2nd ed reading addison wesley lccn 76087042 external links edit wikimedia commons has media related to cones weisstein eric w cone mathworld weisstein eric w double cone mathworld weisstein eric w generalized cone mathworld an interactive spinning cone from maths is fun paper model cone lateral surface area of an oblique cone cut a cone an interactive demonstration of the intersection of a cone with a plane retrieved from https en wikipedia org w index php title cone oldid 1111178425 categories elementary shapes surfaces hidden categories harv and sfn no target errors cs1 julian gregorian uncertainty articles with short description short description matches wikidata commons category link is on wikidata navigation menu personal tools not logged in talk 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