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s a circle the cylinder is called a circular cylinder in some elementary treatments a cylinder always means a circular cylinder 2 the 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 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 axis of the cylinder and it passes through the centers of the two bases a right circular cylinder with radius r and height h right circular cylinders edit 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 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 3 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 edit cylindric sections edit cylindric section a cylindric section is the intersection of a cylinder s surface with a plane they are in general curves and are special types of plane sections the cylindric section by a plane that contains two elements of a cylinder is a parallelogram 4 such a cylindric section of a right cylinder is a rectangle 4 a cylindric section in which the intersecting plane intersects and is perpendicular to all the elements of the cylinder is called a right section 5 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 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 6 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 e of the cylindric section and semi major axis a of the cylindric section depend on the radius of the cylinder r and the angle α between the secant plane and cylinder axis in the following way e cos α displaystyle e cos alpha a r sin α displaystyle a frac r sin alpha volume edit if the base of a circular cylinder has a radius r and the cylinder has height h then its volume is given by v π r 2 h this formula holds whether or not the cylinder is a right cylinder 7 this formula may be established by using cavalieri s principle a solid elliptic cylinder with the semi axes a and b for the base ellipse and height 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 axis a semi minor axis b and height h has a volume v ah where a is the area of the base ellipse π ab this result for right elliptic cylinders can also be obtained by integration where the axis of the cylinder is taken as the positive x axis and a x a the area of each elliptic cross section thus v 0 h a x d x 0 h π a b d x π a b 0 h d x π a b h displaystyle v int _ 0 h a x dx int _ 0 h pi abdx pi ab int _ 0 h dx pi abh using cylindrical coordinates the volume of a right circular cylinder can be calculated by integration over 0 h 0 2 π 0 r s d s d ϕ d z displaystyle int _ 0 h int _ 0 2 pi int _ 0 r s ds d phi dz π r 2 h displaystyle pi r 2 h surface area edit having radius r and altitude height 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 π r 2 the area of the bottom base π r 2 the area of the side 2π rh the area of the top and bottom bases is the same and is called the base area b the area of the side is known as the lateral area l an open cylinder does not include either top or bottom elements and therefore has surface area lateral area l 2π rh 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 a l 2 b 2π rh 2π r 2 2π r h r π d r h where 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 h 2 r equivalently for a given surface area the right circular cylinder with the largest volume has h 2 r that is the cylinder fits snugly in a cube of side length altitude diameter of base circle 8 the lateral area l of a circular cylinder which need not be a right cylinder is more generally given by l e p where e is the length of an element and p is the perimeter of a right section of the cylinder 9 this produces the previous formula for lateral area when the cylinder is a right circular cylinder hollow cylinder right circular hollow cylinder cylindrical shell edit a right circular hollow cylinder or cylindrical shell is a three dimensional region bounded by two right circular cylinders having the same axis and two parallel annular bases perpendicular to the cylinders common axis as in the diagram let the height be h internal radius r and external radius r the volume is given by v π r 2 r 2 h 2 π r r 2 h r r displaystyle v pi r 2 r 2 h 2 pi left frac r r 2 right h r r thus the volume of a cylindrical shell equals 2 π average radius altitude thickness 10 the surface area including the top and bottom is given by a 2 π r r h 2 π r 2 r 2 displaystyle a 2 pi r r h 2 pi r 2 r 2 cylindrical shells are used in a common integration technique for finding volumes of solids of revolution 11 on the sphere and cylinder edit a sphere has 2 3 the volume and surface area of its circumscribing cylinder including its bases main article on the sphere and cylinder in the treatise by this name written c 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 circumscribed right circular cylinder of the same height and diameter the sphere has a volume two thirds that of the circumscribed cylinder and a surface area 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 r is 4 3 π r 3 2 3 2 π r 3 the surface area of this sphere is 4 π r 2 2 3 6 π r 2 a sculpted sphere and cylinder were placed on the tomb of archimedes at his request cylindrical surfaces edit 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 12 such cylinders have at times been referred to as generalized cylinders through each point of a generalized cylinder there passes a unique line that is contained in the cylinder 13 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 surfaces 14 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 f x y z a x 2 b y 2 c z 2 d x e y g z h 0 displaystyle f x y z ax 2 by 2 cz 2 dx ey gz h 0 with the coefficients being real numbers and not all of a b and 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 z does not appear and the general equation of this type of degenerate quadric can be written as 15 a x d 2 a 2 b y e 2 b 2 ρ displaystyle a left x frac d 2a right 2 b left y frac e 2b right 2 rho where ρ h d 2 4 a e 2 4 b displaystyle rho h frac d 2 4a frac e 2 4b elliptic cylinder edit if ab 0 this is the equation of an elliptic cylinder 15 further simplification can be obtained by translation of axes and scalar multiplication if ρ displaystyle rho has the same sign as the coefficients a and b then the equation of an elliptic cylinder may be rewritten in cartesian coordinates as x a 2 y b 2 1 displaystyle left frac x a right 2 left frac y b right 2 1 this equation of an elliptic cylinder is a generalization of the equation of the ordinary circular cylinder a b elliptic cylinders are also known as cylindroids but that name is ambiguous as it can also refer to the plücker conoid if ρ displaystyle rho has a different sign than the coefficients we obtain the imaginary elliptic cylinders x a 2 y b 2 1 displaystyle left frac x a right 2 left frac y b right 2 1 which have no real points on them ρ 0 displaystyle rho 0 gives a single real point hyperbolic cylinder edit if a and b have different signs and ρ 0 displaystyle rho neq 0 we obtain the hyperbolic cylinders whose equations may be rewritten as x a 2 y b 2 1 displaystyle left frac x a right 2 left frac y b right 2 1 parabolic cylinder edit finally if ab 0 assume without loss of generality that b 0 and a 1 to obtain the parabolic cylinders with equations that can be written as 16 x 2 2 a y 0 displaystyle x 2 2a y 0 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 projective geometry edit in projective geometry a cylinder is simply a cone whose 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 17 this concept is useful when considering degenerate conics which may include the cylindrical conics prisms edit 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 n gonal prism where n approaches infinity the connection is very strong and many older texts treat 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 18 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 of a bicone as an infinite sided bipyramid family of uniform n gonal prisms v t e prism name digonal prism trigonal triangular prism tetragonal square prism pentagonal prism hexagonal prism heptagonal prism octagonal prism enneagonal prism decagonal prism hendecagonal prism dodecagonal prism apeirogonal prism polyhedron image spherical tiling image plane tiling image vertex config 2 4 4 3 4 4 4 4 4 5 4 4 6 4 4 7 4 4 8 4 4 9 4 4 10 4 4 11 4 4 12 4 4 4 4 coxeter diagram see also edit list of shapes steinmetz solid the intersection of two or three perpendicular cylinders notes edit κύλινδρος archived 2013 07 30 at the wayback machine henry george liddell robert scott a greek english lexicon on perseus jacobs harold r 1974 geometry w h freeman and co p 607 isbn 0 7167 0456 0 swokowski 1983 p 283 a b wentworth smith 1913 p 354 wentworth smith 1913 p 357 mathworld cylindric section archived from the original on 2008 04 23 wentworth smith 1913 p 359 lax peter d terrell maria shea 2013 calculus with applications undergraduate texts in mathematics springer p 178 isbn 9781461479468 archived from the original on 2018 02 06 wentworth smith 1913 p 358 swokowski 1983 p 292 swokowski 1983 p 291 albert 2016 p 43 albert 2016 p 49 brannan david a esplen matthew f gray jeremy j 1999 geometry cambridge university press p 34 isbn 978 0 521 59787 6 a b albert 2016 p 74 albert 2016 p 75 pedoe dan 1988 1970 geometry a comprehensive course dover p 398 isbn 0 486 65812 0 slaught h e lennes n j 1919 solid geometry with problems and applications pdf revised ed allyn and bacon pp 79 81 archived pdf from the original on 2013 03 06 references edit albert abraham adrian 2016 1949 solid analytic geometry dover isbn 978 0 486 81026 3 swokowski earl w 1983 calculus with analytic geometry alternate ed prindle weber schmidt isbn 0 87150 341 7 wentworth george smith david eugene 1913 plane and solid geometry ginn and co external links edit wikimedia commons has media related to cylinder geometry wikisource has the text of the 1911 encyclopædia britannica article cylinder look up cylinder in wiktionary the free dictionary weisstein eric w cylinder mathworld surface area of a cylinder at mathguide volume of a cylinder at mathguide v t e compact topological surfaces and their immersions in 3d without boundary orientable sphere genus 0 torus genus 1 number 8 genus 2 pretzel genus 3 non orientable real projective plane genus 1 boy s surface roman surface klein bottle genus 2 dyck s surface genus 3 with boundary disk semisphere ribbon annulus cylinder möbius strip cross cap sphere with three holes related notion...