View source for Cylinder

{{Short description|Three-dimensional solid}}
{{Other uses}}
{{Infobox polyhedron
| name          = Cylinder
| image         = File:Cylinder.svg
| caption       = A circular right cylinder of height ''h'' and diameter ''d''=2''r''
| euler         = 2
| symmetry      = [[Orthogonal group|{{math|O(2)×O(1)}}]]
| surface_area  = {{math|2πr(r + h)}}
| volume        = {{math|πr<sup>2</sup>h}}
| type          = [[Smooth surface]]<br />[[Algebraic surface]]
}}

A '''cylinder''' ({{etymology|grc|''{{wikt-lang|grc|κύλινδρος}}'' ({{grc-transl|κύλινδρος}})|roller, tumbler}})<ref>[https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dku%2Flindros κύλινδρος] {{webarchive|url=https://web.archive.org/web/20130730214825/http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dku%2Flindros |date=2013-07-30 }}, Henry George Liddell, Robert Scott, ''A Greek-English Lexicon'', on Perseus</ref> has traditionally been a [[Solid geometry|three-dimensional solid]], one of the most basic of [[curvilinear]] geometric [[shape]]s. In [[elementary geometry]], it is considered a [[Prism (geometry)|prism]] with a [[circle]] as its base.

A cylinder may also be defined as an [[infinite set|infinite]] curvilinear [[surface (mathematics)|surface]] in various modern branches of geometry and [[topology]]. The shift in the basic meaning—solid versus surface (as in a solid [[ball (mathematics)|ball]] versus [[sphere]] surface)—has created some ambiguity with terminology. The two concepts may be distinguished by referring to '''solid cylinders''' and '''cylindrical surfaces'''. In the literature the unadorned term "cylinder" could refer to either of these or to an even more specialized object, the ''[[right circular cylinder]]''.

==Types==
The definitions and results in this section are taken from the 1913 text ''Plane and Solid Geometry'' by [[George A. Wentworth]] and David Eugene Smith {{harv|Wentworth|Smith|1913}}.

A ''{{dfn|cylindrical surface}}'' is a [[Surface (mathematics)|surface]] consisting of all the points on all the lines which are [[parallel lines|parallel]] to a given line and which pass through a fixed [[plane curve]] in a plane not parallel to the given line. Any line in this family of parallel lines is called an ''element'' of the cylindrical surface. From a [[kinematics]] point of view, given a plane curve, called the ''directrix'', a cylindrical surface is that surface traced out by a line, called the ''generatrix'', not in the plane of the directrix, moving parallel to itself and always passing through the directrix. Any particular position of the generatrix is an element of the cylindrical surface.
[[File:Cylinders.svg|thumb|200px|A right and an oblique circular cylinder]]
A [[Solid geometry|solid]] bounded by a cylindrical surface and two [[parallel planes]] is called a (solid) ''{{dfn|cylinder}}''. The line segments determined by an element of the cylindrical surface between the two parallel planes is called an ''element of the cylinder''. All the elements of a cylinder have equal lengths. The region bounded by the cylindrical surface in either of the parallel planes is called a ''{{dfn|base}}'' of the cylinder. The two bases of a cylinder are [[congruence (geometry)|congruent]] figures. If the elements of the cylinder are perpendicular to the planes containing the bases, the cylinder is a ''{{dfn|right cylinder}}'', otherwise it is called an ''{{dfn|oblique cylinder}}''. If the bases are [[Disk (mathematics)|disks]] (regions whose boundary is a [[Circle (mathematics)|circle]]) the cylinder is called a ''{{dfn|circular cylinder}}''. In some elementary treatments, a cylinder always means a circular cylinder.<ref>{{citation |first=Harold R. |last=Jacobs |title=Geometry |year=1974 |publisher=W. H. Freeman and Co. |isbn=0-7167-0456-0 |page=607 }}</ref>
An ''{{dfn|open cylinder}}'' is a cylindrical surface without the bases.

The ''{{dfn|height}}'' (or altitude) of a cylinder is the [[perpendicular]] distance between its bases.

The cylinder obtained by rotating a [[line segment]] about a fixed line that it is parallel to is a ''{{dfn|cylinder of revolution}}''. A cylinder of revolution is a right circular cylinder. The height of a cylinder of revolution is the length of the generating line segment. The line that the segment is revolved about is called the ''{{dfn|axis}}'' of the cylinder and it passes through the centers of the two bases.
[[File:Circular cylinder rh.svg|thumb|180px|A right circular cylinder with radius {{math|''r''}} and height {{math|''h''}}]]

===Right circular cylinders===
{{main|Right circular cylinder}}
The bare term ''cylinder'' often refers to a solid cylinder with circular ends perpendicular to the axis, that is, a right circular cylinder, as shown in the figure. The cylindrical surface without the ends is called an ''{{dfn|open cylinder}}''. The formulae for the [[surface area]] and the [[volume]] of a right circular cylinder have been known from early antiquity.

A right circular cylinder can also be thought of as the [[solid of revolution]] generated by rotating a rectangle about one of its sides. These cylinders are used in an integration technique (the "disk method") for obtaining volumes of solids of revolution.{{sfn|Swokowski|1983|p=283}}

A tall and thin ''needle cylinder'' has a height much greater than its diameter, whereas a short and wide ''disk cylinder'' has a diameter much greater than its height.

==Properties==

===Cylindric sections===
[[Image:Cylindric section.svg|thumb|left|120px|Cylindric section]]

A cylindric section is the intersection of a cylinder's surface with a [[Plane (geometry)|plane]]. They are, in general, curves and are special types of [[cross section (geometry)|''plane sections'']]. The cylindric section by a plane that contains two elements of a cylinder is a [[parallelogram]].{{sfn|Wentworth|Smith|1913|p=354}} Such a cylindric section of a right cylinder is a [[rectangle]].{{sfn|Wentworth|Smith|1913|p=354}}

A cylindric section in which the intersecting plane intersects and is perpendicular to all the elements of the cylinder is called a ''{{dfn|right section}}''.{{sfn|Wentworth|Smith|1913|p=357}} If a right section of a cylinder is a circle then the cylinder is a circular cylinder. In more generality, if a right section of a cylinder is a [[conic section]] (parabola, ellipse, hyperbola) then the solid cylinder is said to be parabolic, elliptic and hyperbolic, respectively.
[[File:Blue cut-cylinder.gif|thumb|Cylindric sections of a right circular cylinder]]
For a right circular cylinder, there are several ways in which planes can meet a cylinder. First, planes that intersect a base in at most one point. A plane is tangent to the cylinder if it meets the cylinder in a single element.  The right sections are circles and all other planes intersect the cylindrical surface in an [[ellipse]].<ref>{{cite web |mode=cs2 |title=Cylindric section |website= [[MathWorld]] |url=http://mathworld.wolfram.com/CylindricSection.html }}</ref> If a plane intersects a base of the cylinder in exactly two points then the line segment joining these points is part of the cylindric section. If such a plane contains two elements, it has a rectangle as a cylindric section, otherwise the sides of the cylindric section are portions of an ellipse. Finally, if a plane contains more than two points of a base, it contains the entire base and the cylindric section is a circle.

In the case of a right circular cylinder with a cylindric section that is an ellipse, the [[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 {{math|''α''}} between the secant plane and cylinder axis, in the following way:
<math display="block">\begin{align}
e &= \cos\alpha, \\[1ex]
a &= \frac{r}{\sin\alpha}.
\end{align}</math>

===Volume===
If the base of a circular cylinder has a [[radius]]  {{math|''r''}} and the cylinder has height {{mvar|h}}, then its [[volume]] is given by
<math display=block>V = \pi r^2h</math>

This formula holds whether or not the cylinder is a right cylinder.{{sfn|Wentworth|Smith|1913|p=359}}

This formula may be established by using [[Cavalieri's principle]].
[[File:Elliptic cylinder abh.svg|thumb|A solid elliptic right cylinder with the semi-axes {{math|''a''}} and {{math|''b''}} for the base ellipse and height {{math|''h''}}]]
In more generality, by the same principle, the volume of any cylinder is the product of the area of a base and the height. For example, an elliptic cylinder with a base having [[Semi-major and semi-minor axes|semi-major axis]] {{mvar|a}}, semi-minor axis {{mvar|b}} and height {{mvar|h}} has a volume {{math|1=''V'' = ''Ah''}}, where {{mvar|A}} is the area of the base ellipse (= {{math|{{pi}}''ab''}}). This result for right elliptic cylinders can also be obtained by integration, where the axis of the cylinder is taken as the positive {{mvar|x}}-axis and {{math|1=''A''(''x'') = ''A''}} the area of each elliptic cross-section, thus:
<math display=block>V = \int_0^h A(x) dx = \int_0^h \pi ab dx = \pi ab \int_0^h dx = \pi a b h.</math>

Using [[cylindrical coordinates]], the volume of a right circular cylinder can be calculated by integration
<math display=block>\begin{align}
V &= \int_0^h \int_0^{2\pi} \int_0^r s \,\, ds \, d\phi \, dz \\[5mu]
&= \pi\,r^2\,h.
\end{align}</math>

===Surface area===
Having radius {{math|''r''}} and altitude (height) {{mvar|h}}, the [[surface area]] of a right circular cylinder, oriented so that its axis is vertical, consists of three parts:

* the area of the top base: {{math|π''r''<sup>2</sup>}}
* the area of the bottom base: {{math|π''r''<sup>2</sup>}}
* the area of the side: {{math|2π''rh''}}

The area of the top and bottom bases is the same, and is called the ''base area'', {{math|''B''}}. The area of the side is known as the ''{{dfn|lateral area}}'', {{math|''L''}}.

An ''open cylinder'' does not include either top or bottom elements, and therefore has surface area (lateral area)
<math display=block>L = 2 \pi r h</math>

The surface area of the solid right circular cylinder is made up the sum of all three components: top, bottom and side. Its surface area is therefore
<math display=block>A = L + 2B = 2\pi rh + 2\pi r^2 = 2 \pi r (h + r) = \pi d (r + h)</math>
where {{math|1=''d'' = 2''r''}} is the [[diameter]] of the circular top or bottom.

For a given volume, the right circular cylinder with the smallest surface area has {{math|1=''h'' = 2''r''}}. Equivalently, for a given surface area, the right circular cylinder with the largest volume has {{math|1=''h'' = 2''r''}}, that is, the cylinder fits snugly in a cube of side length = altitude ( = diameter of base circle).<ref>{{citation |title=Calculus With Applications |first1=Peter D. |last1=Lax |author1-link=Peter Lax |first2=Maria Shea |last2=Terrell |publisher=Springer |year=2013 |isbn=9781461479468 |page=178 |url=https://books.google.com/books?id=dDq3BAAAQBAJ&pg=PA178 }}.</ref>

The lateral area, {{mvar|L}}, of a circular cylinder, which need not be a right cylinder, is more generally given by
<math display=block>L = e \times p,</math>
where {{mvar|e}} is the length of an element and {{mvar|p}} is the perimeter of a right section of the cylinder.{{sfn|Wentworth|Smith|1913|p=358}} This produces the previous formula for lateral area when the cylinder is a right circular cylinder.
[[File:Zylinder-rohr-s.svg|thumb|180px|Hollow cylinder]]

=== Right circular hollow cylinder (cylindrical shell)===

A ''right circular hollow cylinder'' (or ''{{dfn|cylindrical shell}}'') is a three-dimensional region bounded by two right circular cylinders having the same axis and two parallel [[Annulus (mathematics)|annular]] bases perpendicular to the cylinders' common axis, as in the diagram.

Let the height be {{math|''h''}}, internal radius {{math|''r''}}, and external radius {{math|''R''}}.  The volume is given by
<math display=block>
V = \pi \left( R ^2 - r ^2 \right) h
= 2 \pi \left ( \frac{R + r}{2} \right) h (R - r).
</math>
Thus, the volume of a cylindrical shell equals {{nobr|2{{pi}}&thinsp;×}}&thinsp;{{nobr|average radius&thinsp;×}}&thinsp;{{nobr|altitude&thinsp;×}}&thinsp;thickness.{{sfn|Swokowski|1983|p=292}}

The surface area, including the top and bottom, is given by
<math display=block> A = 2 \pi \left( R + r \right) h + 2 \pi \left( R^2  - r^2 \right). </math>
Cylindrical shells are used in a common integration technique for finding volumes of solids of revolution.{{sfn|Swokowski|1983|p=291}}

===''On the Sphere and Cylinder''===

[[File:Esfera Arquímedes.svg|thumb|right|A sphere has 2/3 the volume and surface area of its circumscribing cylinder including its bases]]
{{main|On the Sphere and Cylinder}}
In the treatise by this name, written {{Circa|225 BCE}}, [[Archimedes]] obtained the result of which he was most proud, namely obtaining the formulas for the volume and surface area of a [[sphere]] by exploiting the relationship between a sphere and its [[circumscribe]]d [[right circular cylinder]] of the same height and [[diameter]]. The sphere has a volume {{nowrap|two-thirds}} that of the circumscribed cylinder and a surface area  {{nowrap|two-thirds}} that of the cylinder (including the bases). Since the values for the cylinder were already known, he obtained, for the first time, the corresponding values for the sphere. The volume of a sphere of radius {{mvar|r}} is {{math|1={{sfrac|4|3}}{{pi}}''r''<sup>3</sup> = {{sfrac|2|3}} (2{{pi}}''r''<sup>3</sup>)}}. The surface area of this sphere is {{math|1=4{{pi}}''r''<sup>2</sup> = {{sfrac|2|3}} (6{{pi}}''r''<sup>2</sup>)}}. A sculpted sphere and cylinder were placed on the tomb of Archimedes at his request.

==Cylindrical surfaces==
{{anchor|elliptic cylinder|parabolic cylinder|hyperbolic cylinder}}
In some areas of geometry and topology the term ''cylinder'' refers to what has been called a '''cylindrical surface'''. A cylinder is defined as a surface consisting of all the points on all the lines which are parallel to a given line and which pass through a fixed plane curve in a plane not parallel to the given line.{{sfn|Albert|2016|p=43}} Such cylinders have, at times, been referred to as ''{{dfn|generalized cylinders}}''. Through each point of a generalized cylinder there passes a unique line that is contained in the cylinder.{{sfn|Albert|2016|p=49}} Thus, this definition may be rephrased to say that a cylinder is any [[ruled surface]] spanned by a one-parameter family of parallel lines.

A cylinder having a right section that is an [[ellipse]], [[parabola]], or [[hyperbola]]  is called an '''elliptic cylinder''', '''parabolic cylinder''' and '''hyperbolic cylinder''', respectively. These are degenerate [[quadric surface]]s.<ref>{{citation |first1=David A. |last1=Brannan |first2=Matthew F. |last2=Esplen |first3=Jeremy J. |last3=Gray |title=Geometry |year=1999 |publisher=Cambridge University Press |isbn=978-0-521-59787-6 |page=34 }}</ref>
[[File:Zylinder-parabol-s.svg|thumb|120px|Parabolic cylinder]]
When the principal axes of a quadric are aligned with the reference frame (always possible for a quadric), a general equation of the quadric in three dimensions is given by
<math display=block>f(x,y,z)=Ax^2 + By^2 + C z^2 + Dx + Ey + Gz + H = 0,</math>
with the coefficients being [[real number]]s and not all of {{mvar|A}}, {{mvar|B}} and {{mvar|C}} being 0. If at least one variable does not appear in the equation, then the quadric is degenerate. If one variable is missing, we may assume by an appropriate [[rotation of axes]] that the variable {{mvar|z}} does not appear and the general equation of this type of degenerate quadric can be written as{{sfn|Albert|2016|p=74}}
<math display=block>A \left ( x + \frac{D}{2A} \right )^2 + B \left(y + \frac{E}{2B} \right)^2 = \rho,</math>
where
<math display=block>\rho = -H + \frac{D^2}{4A} + \frac{E^2}{4B}.</math>

=== Elliptic cylinder ===
If {{math|''AB'' > 0}} this is the equation of an ''elliptic cylinder''.{{sfn|Albert|2016|p=74}} Further simplification can be obtained by [[translation of axes]] and scalar multiplication. If <math>\rho</math> has the same sign as the coefficients {{mvar|A}} and {{mvar|B}}, then the equation of an elliptic cylinder may be rewritten in [[Cartesian coordinates]] as:
<math display=block>\left(\frac{x}{a}\right)^2+ \left(\frac{y}{b}\right)^2 = 1.</math>
This equation of an elliptic cylinder is a generalization of the equation 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]].

If <math>\rho</math> has a different sign than the coefficients, we obtain the ''imaginary elliptic cylinders'':
<math display=block>\left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = -1,</math>
which have no real points on them. (<math>\rho = 0</math> gives a single real point.)

=== Hyperbolic cylinder ===
If {{mvar|A}} and {{mvar|B}} have different signs and <math>\rho \neq 0</math>, we obtain the ''hyperbolic cylinders'', whose equations may be rewritten as:
<math display=block>\left(\frac{x}{a}\right)^2 - \left(\frac{y}{b}\right)^2 = 1.</math>

=== Parabolic cylinder ===
Finally, if {{math|1=''AB'' = 0}} assume, [[without loss of generality]], that {{math|1=''B'' = 0}} and {{math|1=''A'' = 1}} to obtain the ''parabolic cylinders'' with equations that can be written as:{{sfn|Albert|2016|p=75}}
<math display=block> x^2 + 2 a y = 0 .</math>

[[File:(Texas Gulf Sulphur Company) (10428629273).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.]]

==Projective geometry==
In [[projective geometry]], a cylinder is simply a [[cone (geometry)|cone]] whose [[apex (geometry)|apex]] (vertex) lies on the [[plane at infinity]]. If the cone is a quadratic cone, the plane at infinity (which passes through the vertex) can intersect the cone at two real lines, a single real line (actually a coincident pair of lines), or only at the vertex. These cases give rise to the hyperbolic, parabolic or elliptic cylinders respectively.<ref>{{citation |first=Dan |last=Pedoe |title=Geometry a Comprehensive Course |year=1988 |orig-year=1970 |publisher=Dover |isbn=0-486-65812-0 |page=398}}</ref>

This concept is useful when considering [[degenerate conic]]s, which may include the cylindrical conics.

==Prisms==

[[File:TychoBrahePlanetarium-Copenhagen.jpg|thumb|[[Tycho Brahe Planetarium]] building, Copenhagen, is an example of a truncated cylinder]]

A ''solid circular cylinder'' can be seen as the limiting case of a [[regular polygon|{{mvar|n}}-gonal]] prism where {{math|''n''}} approaches [[infinity]]. The connection is very strong and many older texts treat [[Prism (geometry)|prisms]] and cylinders simultaneously. Formulas for surface area and volume are derived from the corresponding formulas for prisms by using inscribed and circumscribed prisms and then letting the number of sides of the prism increase without bound.<ref>{{citation |first1=H.E. |last1=Slaught |author-link=Herbert Ellsworth Slaught |first2=N.J. |last2=Lennes |title=Solid Geometry with Problems and Applications |edition=Rev. |year=1919 |publisher=Allyn and Bacon |url=http://www.gutenberg.org/files/29807/29807-pdf.pdf |pages=79–81}}</ref> One reason for the early emphasis (and sometimes exclusive treatment) on circular cylinders is that a circular base is the only type of geometric figure for which this technique works with the use of only elementary considerations (no appeal to calculus or more advanced mathematics). Terminology about prisms and cylinders is identical. Thus, for example, since a ''truncated prism'' is a prism whose bases do not lie in parallel planes, a solid cylinder whose bases do not lie in parallel planes would be called a ''truncated cylinder''.

From a polyhedral viewpoint, a cylinder can also be seen as a [[dual polyhedron|dual]] of a [[bicone]] as an infinite-sided [[bipyramid]].

{{UniformPrisms}}

==See also==
*[[Lists of shapes]]
*[[Steinmetz solid]], the intersection of two or three perpendicular cylinders

{{clear}}

==Notes==
{{reflist |25em }}

==References==
{{refbegin}}
*{{citation|first=Abraham Adrian|last=Albert|title=Solid Analytic Geometry|year=2016|orig-year=1949|publisher=Dover|isbn=978-0-486-81026-3}}
*{{citation|first=Earl W.|last=Swokowski|title=Calculus with Analytic Geometry|edition=Alternate|year=1983|publisher=Prindle, Weber & Schmidt|isbn=0-87150-341-7|url-access=registration|url=https://archive.org/details/calculuswithanal00swok}}
*{{citation|first1=George|last1=Wentworth|first2=David Eugene|last2=Smith|title=Plane and Solid Geometry|year=1913|publisher=Ginn and Co.}}
{{refend}}

==External links==
{{Commons}}
{{Wiktionary|cylinder}}
{{EB1911 poster|Cylinder}}
*{{MathWorld|title=Cylinder|id=Cylinder}}
*[https://web.archive.org/web/20141009113911/http://www.mathguide.com/lessons/SurfaceArea.html#cylinders Surface area of a cylinder] at MATHguide
*[https://web.archive.org/web/20141009114334/http://www.mathguide.com/lessons/Volume.html#cylinders Volume of a cylinder] at MATHguide

{{Compact topological surfaces}}
{{Authority control}}

[[Category:Quadrics]]
[[Category:Elementary shapes]]
[[Category:Euclidean solid geometry]]
[[Category:Surfaces]]