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| [[Image:Circular cylinder rh.svg|thumb|A right circular cylinder with radius ''r'' and height ''h''.]]
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| A '''cylinder''' (from [[Greek language|Greek]] κύλινδρος – ''kulindros'', "roller, tumbler"<ref>[http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dku%2Flindros κύλινδρος], Henry George Liddell, Robert Scott, ''A Greek-English Lexicon'', on Perseus</ref>) is one of the most basic curvilinear geometric shapes, the [[surface]] formed by the points at a fixed distance from a given [[line segment]], the '''axis''' of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder. The [[surface area]] and the [[volume]] of a cylinder have been known since deep antiquity.
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| In [[differential geometry]], a cylinder is defined more broadly as any [[ruled surface]] spanned by a one-parameter family of parallel lines. A cylinder whose [[Cross section (geometry)|cross section]] is an [[ellipse]], [[parabola]], or [[hyperbola]] is called an '''elliptic cylinder''', '''parabolic cylinder''', or '''hyperbolic cylinder''' respectively.
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| The open cylinder is [[Homeomorphism|topologically equivalent]] to both the open [[annulus (mathematics)|annulus]] and the [[punctured plane]].
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| ==Common use==
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| In common use a ''cylinder'' is taken to mean a finite section of a '''''right circular cylinder''''', i.e., the cylinder with the generating lines perpendicular to the bases, with its ends closed to form two circular surfaces, as in the figure (right). If the cylinder has a [[radius]] {{math|''r''}} and length (height) {{mvar|h}}, then its [[volume]] is given by
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| :{{math|1=''V'' = π''r''<sup>2</sup>''h''}}
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| and its [[surface area]] is:
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| * the area of the top {{math|(π''r''<sup>2</sup>) +}}
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| * the area of the bottom {{math|(π''r''<sup>2</sup>) +}}
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| * the area of the side ({{math|2π''rh''}}).
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| Therefore without the top or bottom (lateral area), the surface area is:
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| :{{math|1=''A'' = 2π''rh''}}.
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| With the top and bottom, the surface area is:
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| :{{math|1=''A'' = 2π''r''<sup>2</sup> + 2π''rh'' = 2π''r''(''r'' + ''h'')}}.
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| For a given volume, the cylinder with the smallest surface area has {{math|1=''h'' = 2''r''}}. For a given surface area, the cylinder with the largest volume has {{math|1=''h'' = 2''r''}}, i.e. the cylinder fits snugly in a cube (height = diameter). | |
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| ==Volume==
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| Having a right circular cylinder with a height {{math|''h''}} units and a base of radius {{math|''r''}} units with the coordinate axes chosen so that the origin is at the center of one base and the height is measured along the positive x-axis. A plane section at a distance of {{math|''x''}} units from the origin has an area of {{math|''A''(''x'')}} square units where
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| ::<math>A(x)=\pi r^2</math>
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| or
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| ::<math>A(y)=\pi r^2</math>
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| An element of volume, is a right cylinder of base area {{math|''Aw<sub>i</sub>''}} square units and a thickness of {{math|Δ<sub>''i''</sub>''x''}} units. Thus if {{mvar|V}} cubic units is the volume of the right circular cylinder, by [[Riemann sums]],
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| ::<math>\mathrm{Volume \; of \; cylinder}=\lim_{||\Delta \to 0 ||} \sum_{i=1}^n A(w_i) \Delta_i x</math>
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| :::<math>=\int_{0}^{h} A(y) \, dy</math>
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| :::<math>=\int_{0}^{h} \pi r^2 \, dy</math>
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| :::<math>=\pi\,r^2\,h\,</math>
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| Using cylindrical coordinates, the volume can be calculated by integration over
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| :::<math>=\int_{0}^{h} \int_{0}^{2\pi} \int_{0}^{r} s \,\, ds \, d\phi \, dz</math>
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| :::<math>=\pi\,r^2\,h\,</math>
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| [[File:TychoBrahePlanetarium-Copenhagen.jpg|thumb|left|[[Tycho Brahe Planetarium]] building, its roof being an example of a cylindric section]]
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| ==Surface area==
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| The [[formula]] for finding the surface area of a cylinder is, with ''h'' as height, ''r'' as radius, and ''S'' as surface area is | |
| <math>S=2\pi rh+2\pi r^2</math>
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| Or, with ''B'' as base area and ''L'' as lateral area,
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| <math>S=L+2B</math>
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| ==Cylindric section==
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| [[Image:Cylindric section.svg|thumb|120px|Cylindric section.]]
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| Cylindric sections are the intersections of cylinders with planes. For a right circular cylinder, there are four possibilities. A plane tangent to the cylinder, meets the cylinder in a single straight line. Moved while parallel to itself, the plane either does not intersect the cylinder or intersects it in two parallel lines. All other planes intersect the cylinder in an ellipse or, when they are perpendicular to the axis of the cylinder, in a circle.<ref>{{cite web | title=MathWorld: Cylindric section| url=http://mathworld.wolfram.com/CylindricSection.html}}</ref>
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| [[Eccentricity (mathematics)|Eccentricity]] {{math|''e''}} of the cylindric section and [[semi-major axis]] {{math|''a''}} of the cylindric section depend on the radius of the cylinder {{math|''r''}} and the angle between the secant plane and cylinder axis {{math|''α''}} in the following way:
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| :::<math>e=\cos\alpha\,</math>
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| :::<math>a=\frac{r}{\sin\alpha}\,</math>
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| ==Other types of cylinders==
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| [[Image:Elliptic cylinder abh.svg|thumb|An elliptic cylinder with the half-axes ''a'' and ''b'' for the surface ellipse and the height ''h''.]]
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| An '''elliptic cylinder''' is a [[quadric surface]], with the following equation in [[Cartesian coordinates]]:
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| :<math>\left(\frac{x}{a}\right)^2+ \left(\frac{y}{b}\right)^2 = 1</math>
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| This equation is for an '''elliptic cylinder''', a generalization of the ordinary, '''circular cylinder''' ({{math|1=''a'' = ''b''}}). Elliptic cylinders are also known as '''cylindroids''', but that name is ambiguous, as it can also refer to the [[Plücker conoid]].
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| Even more general than the elliptic cylinder is the '''generalized cylinder''': the [[Cross section (geometry)|cross-section]] can be any curve.
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| The cylinder is a ''degenerate [[quadric]]'' because at least one of the coordinates (in this case {{math|z}}) does not appear in the equation.
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| An '''oblique cylinder''' has the top and bottom surfaces displaced from one another.
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| There are other more unusual types of cylinders. These are the ''imaginary elliptic cylinders'':
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| :<math>\left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = -1</math>
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| the ''hyperbolic cylinder'':
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| :<math>\left(\frac{x}{a}\right)^2 - \left(\frac{y}{b}\right)^2 = 1</math>
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| and the ''parabolic cylinder'':
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| :<math> {x}^2+2a{y}=0 \,</math>
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| ==About an arbitrary axis==
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| Consider an infinite cylinder whose axis lies along the vector
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| :<math> \overrightarrow{v} = (\alpha, \beta, \gamma) \,</math>
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| We make use of [[Spherical_coordinate_system|spherical coordinates]]:
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| :<math>\rho^2=\alpha^2+\beta^2+\gamma^2\,</math>
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| :<math>\theta=\arctan\left(\frac{\beta}{\alpha}\right)</math>
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| :<math>\phi=\arcsin\left(\frac{\gamma}{\rho}\right)</math>
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| These variables can be used to define A and B, the orthogonal vectors that form the basis for the cylinder:
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| <math>A=-x\sin(\theta)+y\cos(\theta)cos(\phi)+z\cos(\theta)\sin(\phi)</math>
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| <math>B = -y\sin(\phi)+z\cos(\phi)</math>
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| With these defined, we may use the familiar formula for a cylinder:
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| :<math> A^2 + B^2 = R^2 \,</math>
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| where ''R'' is the radius of the cylinder.
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| These results are usually derived using [[Rotation_matrix|rotation matrices]].
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| == Projective geometry ==
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| [[File:Canterra Tower-Calgary.JPG|thumb|In [[projective geometry]], a cylinder is simply a cone whose [[apex (geometry)|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 (geometry)|cone]] whose [[apex (geometry)|apex]] is at infinity.
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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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| *[[Steinmetz solid]], the intersection of two or three perpendicular cylinders
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| *[[Cylindrical coordinates]], the cylindrical coordinate system
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| == Related polyhedra ==
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| A ''cylinder'' can be seen as a [[polyhedron|polyhedral]] limiting case of an [[regular polygon|n-gonal]] [[Prism (geometry)|prism]] where ''n'' approaches [[infinity]]. It can also be seen as a [[dual polyhedron|dual]] of a [[bicone]] as an infinite-sided [[bipyramid]].
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| {{UniformPrisms}}
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| ==References==
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| {{reflist|2}}
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| ==External links==
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| {{commons|Cylinder (geometry)}}
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| {{Wikisource1911Enc|Cylinder}}
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| {{Wiktionary|cylinder}}
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| *[http://www.mathguide.com/lessons/SurfaceArea.html#cylinders Surface area of a cylinder] at MATHguide
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| *[http://www.mathguide.com/lessons/Volume.html#cylinders Volume of a cylinder] at MATHguide
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| [[Category:Quadrics]]
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| [[Category:Elementary shapes]]
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| [[Category:Euclidean solid geometry]]
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| [[Category:Surfaces]]
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