TheInfoList

A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of
curvilinear In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of spac ...

geometric shapes. Geometrically, it can be considered as a
prism An optical prism is a transparent optics, optical element with flat, polished surfaces that refraction, refract light. At least one surface must be angled—elements with two parallel surfaces are not prisms. The traditional geometrical shape o ...
with a
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre; equivalently it is the curve traced out by a point that moves in a ...

as its base. This traditional view is still used in elementary treatments of geometry, but the advanced mathematical viewpoint has shifted to the
infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (band), a South Korean boy band *''Infinite'' (EP), debut EP of American musi ...
curvilinear
surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...
and this is how a cylinder is now defined in various modern branches of geometry and
topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

. The shift in the basic meaning (solid versus surface) has created some ambiguity with terminology. It is generally hoped that context makes the meaning clear. Both points of view are typically presented and distinguished by referring to ''solid cylinders'' and ''cylindrical surfaces'', but 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 Wentworth and David Eugene Smith . A ' is a
surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...
consisting of all the points on all the lines which are
parallel Parallel may refer to: Computing * Parallel algorithm In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their a ...

to a given line and which pass through a fixed
plane curve In mathematics, a plane curve is a curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of ...
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 Kinematics is a subfield of physics, developed in classical mechanics, that describes the Motion (physics), motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kine ...

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. A
solid Solid is one of the four fundamental states of matter (the others being liquid A liquid is a nearly incompressible fluid In physics, a fluid is a substance that continually Deformation (mechanics), deforms (flows) under an applied ...
bounded by a cylindrical surface and two
parallel planes In geometry, parallel lines are line (geometry), lines in a plane (geometry), plane which do not meet; that is, two straight lines in a plane that do not intersecting lines, intersect at any point are said to be parallel. Colloquially, curves tha ...
is called a (solid) '. 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 ' of the cylinder. The two bases of a cylinder are
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...
figures. If the elements of the cylinder are perpendicular to the planes containing the bases, the cylinder is a ', otherwise it is called an '. If the bases are disks (regions whose boundary is a
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre; equivalently it is the curve traced out by a point that moves in a ...
) the cylinder is called a '. In some elementary treatments, a cylinder always means a circular cylinder. The ' (or altitude) of a cylinder is the
perpendicular In elementary geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with propertie ...

distance between its bases. The cylinder obtained by rotating a
line segment 250px, The geometric definition of a closed line segment: the intersection of all points at or to the right of ''A'' with all points at or to the left of ''B'' In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' ...

about a fixed line that it is parallel to is a '. 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 ' of the cylinder and it passes through the centers of the two bases.

## Right circular cylinders

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 '. The formulae for the
surface area of radius has surface area . The surface area of a Solid geometry, solid object is a measure of the total area that the Surface (mathematics), surface of the object occupies. The mathematical definition of surface area in the presence of curved ...

and the
volume Volume is a scalar quantity (physics), scalar quantity expressing the amount of three-dimensional space enclosed by a closed surface. For example, the space that a substance (solid, liquid, gas, or Plasma (physics), plasma) or 3D shape occupies ...

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 In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

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.

# Properties

## Cylindric sections

left, 120px, Cylindric section A cylindric section is the intersection of a cylinder's surface with a
plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons''), a location in the multiverse *Plane (Magic: Th ...
. 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 In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method con ...

. Such a cylindric section of a right cylinder is a
rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a para ...

. A cylindric section in which the intersecting plane intersects and is perpendicular to all the elements of the cylinder is called a '. 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
(parabola, ellipse, hyperbola) then the solid cylinder is said to be parabolic, elliptic and hyperbolic, respectively. 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 In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a circle, which is the sp ...

. 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 Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off- center, in geometry * Eccentricity (graph theory) of a ...
of the cylindric section and
semi-major axis In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...
of the cylindric section depend on the radius of the cylinder and the angle between the secant plane and cylinder axis, in the following way: :::$e=\cos\alpha,$ :::$a=\frac.$

## Volume

If the base of a circular cylinder has a
radius In classical geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties ...

and the cylinder has height , then its
volume Volume is a scalar quantity (physics), scalar quantity expressing the amount of three-dimensional space enclosed by a closed surface. For example, the space that a substance (solid, liquid, gas, or Plasma (physics), plasma) or 3D shape occupies ...

is given by :. This formula holds whether or not the cylinder is a right cylinder. This formula may be established by using
Cavalieri's principle In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...
. 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 In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...
, semi-minor axis and height has a volume , where is the area of the base ellipse (= ). This result for right elliptic cylinders can also be obtained by integration, where the axis of the cylinder is taken as the positive -axis and the area of each elliptic cross-section, thus: :$V=\int_0^h A\left(x\right) dx = \int_0^h \pi ab dx = \pi ab \int_0^h dx = \pi abh.$ Using
cylindrical coordinates 240px, A cylindrical coordinate system with origin , polar axis , and longitudinal axis . The dot is the point with radial distance , angular coordinate , and height . A cylindrical coordinate system is a three-dimensional coordinate system that s ...

, the volume of a right circular cylinder can be calculated by integration over :::$=\int_^ \int_^ \int_^ s \,\, ds \, d\phi \, dz$ :::$=\pi\,r^2\,h.$

## Surface area

Having radius and altitude (height) , the
surface area of radius has surface area . The surface area of a Solid geometry, solid object is a measure of the total area that the Surface (mathematics), surface of the object occupies. The mathematical definition of surface area in the presence of curved ...

of a right circular cylinder, oriented so that its axis is vertical, consists of three parts: * the area of the top base: * the area of the bottom base: * the area of the side: The area of the top and bottom bases is the same, and is called the ''base area'', . The area of the side is known as the ', . An ''open cylinder'' does not include either top or bottom elements, and therefore has surface area (lateral area) :. 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, :, where is the
diameter In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

of the circular top or bottom. For a given volume, the right circular cylinder with the smallest surface area has . Equivalently, for a given surface area, the right circular cylinder with the largest volume has , that is, the cylinder fits snugly in a cube of side length = altitude ( = diameter of base circle). The lateral area, , of a circular cylinder, which need not be a right cylinder, is more generally given by: :, where is the length of an element and is the perimeter of a right section of the cylinder. This produces the previous formula for lateral area when the cylinder is a right circular cylinder.

## Right circular hollow cylinder (cylindrical shell)

A ''right circular hollow cylinder'' (or ') 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 , internal radius , and external radius . The volume is given by :$V = \pi \left( R ^ - r ^ \right) h = 2\pi \left \left( \frac \right\right) h \left(R - r\right).$. Thus, the volume of a cylindrical shell equals 2(average radius)(altitude)(thickness). The surface area, including the top and bottom, is given by :$A = 2 \pi \left( R + r \right) h + 2 \pi \left( R^2 - r^2 \right).$. Cylindrical shells are used in a common integration technique for finding volumes of solids of revolution.

## ''On the Sphere and Cylinder''

In the treatise by this name, written c. 225 BCE,
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its popula ...

obtained the result of which he was most proud, namely obtaining the formulas for the volume and surface area of a
sphere A sphere (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 m ...

by exploiting the relationship between a sphere and its
circumscribe In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...
d right circular cylinder of the same height and
diameter In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

. The sphere has a volume that of the circumscribed cylinder and a surface area 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 is . The surface area of this sphere is . A sculpted sphere and cylinder were placed on the tomb of Archimedes at his request.

# Cylindrical surfaces

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. Such cylinders have, at times, been referred to as '. Through each point of a generalized cylinder there passes a unique line that is contained in the cylinder. Thus, this definition may be rephrased to say that a cylinder is any
ruled surface In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...
spanned by a one-parameter family of parallel lines. A cylinder having a right section that is an
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a circle, which is the sp ...

,
parabola The parabola is a member of the family of conic sections. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

, or
hyperbola File:Hyperbel-def-ass-e.svg, 300px, Hyperbola (red): features In mathematics, a hyperbola () (adjective form hyperbolic, ) (plural ''hyperbolas'', or ''hyperbolae'' ()) is a type of smooth function, smooth plane curve, curve lying in a plane, defi ...

is called an elliptic cylinder, parabolic cylinder and hyperbolic cylinder, respectively. These are degenerate
quadric surface In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims ...
s. 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 :$f\left(x,y,z\right)=Ax^2 + By^2 + Cz^2 + Dx + Ey + Gz + H = 0,$ with the coefficients being
real number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s and not all of , and 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 does not appear and the general equation of this type of degenerate quadric can be written as :$A \left \left( x + \frac \right \right)^2 + B \left\left(y + \frac \right\right)^2 = \rho,$ where :$\rho = -H + \frac + \frac.$

## Elliptic cylinder

If this is the equation of an ''elliptic cylinder''. Further simplification can be obtained by translation of axes and scalar multiplication. If $\rho$ has the same sign as the coefficients and , then the equation of an elliptic cylinder may be rewritten in
Cartesian coordinates A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

as: :$\left\left(\frac\right\right)^2+ \left\left(\frac\right\right)^2 = 1.$ This equation of an elliptic cylinder is a generalization of the equation of the ordinary, ''circular cylinder'' (). Elliptic cylinders are also known as ''cylindroids'', but that name is ambiguous, as it can also refer to the Plücker conoid. If $\rho$ has a different sign than the coefficients, we obtain the ''imaginary elliptic cylinders'': :$\left\left(\frac\right\right)^2 + \left\left(\frac\right\right)^2 = -1,$ which have no real points on them. ($\rho = 0$ gives a single real point.)

## Hyperbolic cylinder

If and have different signs and $\rho \neq 0$, we obtain the ''hyperbolic cylinders'', whose equations may be rewritten as: :$\left\left(\frac\right\right)^2 - \left\left(\frac\right\right)^2 = 1.$

## Parabolic cylinder

Finally, if assume,
without loss of generality ''Without loss of generality'' (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as ''without any loss of generality'' or ''with no loss of generality'') is a frequently used expression in mathematics Mathematics (from Ancie ...
, that and to obtain the ''parabolic cylinders'' with equations that can be written as: :$^2+2a=0 .$

# Projective geometry

In
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, proj ...
, a cylinder is simply 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 c ...

whose
apex Apex may refer to: Arts and media Fictional entities * Apex (comics), a teenaged super villainess in the Marvel Universe * Ape-X, a super-intelligent ape in the Squadron Supreme universe *Apex, a genetically-engineered human population in the TV s ...
(vertex) lies on the
plane at infinity In projective geometry, a plane at infinity is the hyperplane at infinity of a three dimensional projective space or to any Plane (geometry), plane contained in the hyperplane at infinity of any projective space of higher dimension. This article wil ...
. 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. This concept is useful when considering
degenerate conic In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...
s, which may include the cylindrical conics.

# Prisms

A ''solid circular cylinder'' can be seen as the limiting case of a -gonal
prism An optical prism is a transparent optics, optical element with flat, polished surfaces that refraction, refract light. At least one surface must be angled—elements with two parallel surfaces are not prisms. The traditional geometrical shape o ...
where approaches
infinity Infinity is that which is boundless, endless, or larger than any number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything t ...

. 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. 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 Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality ** . . . see more cases in :Duality theories * Dual ...
of a
bicone 100px, right In geometry, a bicone or dicone (from la, bi-, and Greek: ''di-'', both meaning "two") is the three-dimensional surface of revolution of a rhombus around one of its axes of symmetry. Equivalently, a bicone is the surface created b ...

as an infinite-sided
bipyramid A (symmetric) ''n''-gonal bipyramid or dipyramid is a polyhedron In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the old ...
.

*
List of shapes Lists of shapes cover different types of geometric shape A shape is the form of an object or its external boundary, outline, or external surface, as opposed to other properties such as color Color ( American English), or colour ( Comm ...
*
Steinmetz solid In geometry, a Steinmetz solid is the solid body obtained as the intersection of two or three cylinder (geometry), cylinders of equal radius at right angles. Each of the curves of the intersection of two cylinders is an ellipse. The intersection of ...
, the intersection of two or three perpendicular cylinders

* * *