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A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of
curvilinear In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally inv ...
geometric
shape A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type. A plane shape or plane figure is constrained to lie ...
s. In
elementary geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
, it is considered a prism with a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
as its base. A cylinder may also be defined as an infinite curvilinear surface in various modern branches of geometry and
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
. The shift in the basic meaning—solid versus surface (as in
ball A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used f ...
and
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
)—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 Wentworth and David Eugene Smith . A ' is a surface consisting of all the points on all the lines which are
parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster o ...
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 Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a fiel ...
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, gas, and plasma). The molecules in a solid are closely packed together and contain the least amount of kinetic energy. A solid is characterized by structur ...
bounded by a cylindrical surface and two parallel planes 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 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 points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
) 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, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...
distance between its bases. The cylinder obtained by rotating a
line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between i ...
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 The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of ...
and the
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
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, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the '' axis of revolution'') that lies on the same plane. The surface created by this revolution and which bounds the solid is ...
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. 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

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 In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of eq ...
. Such a cylindric section of a right cylinder is a
rectangle In 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 parallelogram contain ...
. 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, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a ...
(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 focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
. 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 of the cylindric section and
semi-major axis In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the lon ...
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, a radius (plural, : radii) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', ...
and the cylinder has height , then its
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
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, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows: * 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that pl ...
. 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, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the lon ...
, 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(x) dx = \int_0^h \pi ab dx = \pi ab \int_0^h dx = \pi abh. Using cylindrical coordinates, 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 The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of ...
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, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid f ...
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 ( R ^ - r ^ ) h = 2\pi \left ( \frac \right) h (R - r). . 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 ( 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.


''On the Sphere and Cylinder''

In the treatise by this name, written c. 225 BCE,
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientis ...
obtained the result of which he was most proud, namely obtaining the formulas for the volume and surface area of a
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
by exploiting the relationship between a sphere and its circumscribed right circular cylinder of the same height and
diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid f ...
. 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 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 focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
,
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descri ...
, or
hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
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 of conic sections ( ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is ...
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(x,y,z)=Ax^2 + By^2 + Cz^2 + Dx + Ey + Gz + H = 0, with the coefficients being
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
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 In mathematics, a rotation of axes in two dimensions is a mapping from an ''xy''-Cartesian coordinate system to an ''x′y′''-Cartesian coordinate system in which the origin is kept fixed and the ''x′'' and ''y′'' axes are ...
that the variable does not appear and the general equation of this type of degenerate quadric can be written as :A \left ( x + \frac \right )^2 + B \left(y + \frac \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 In mathematics, a translation of axes in two dimensions is a mapping from an ''xy''-Cartesian coordinate system to an ''x'y-Cartesian coordinate system in which the ''x axis is parallel to the ''x'' axis and ''k'' units away, and the ''y ...
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 is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
as: :\left(\frac\right)^2+ \left(\frac\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(\frac\right)^2 + \left(\frac\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(\frac\right)^2 - \left(\frac\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. The term is used to indicat ...
, 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, ...
, a cylinder is simply a cone whose apex (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 contained in the hyperplane at infinity of any projective space of higher dimension. This article will be concerned ...
. 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 conics, which may include the cylindrical conics.


Prisms

A ''solid circular cylinder'' can be seen as the limiting case of a
-gonal In mathematics, a polygonal number is a number represented as dots or pebbles arranged in the shape of a regular polygon. The dots are thought of as alphas (units). These are one type of 2-dimensional figurate numbers. Definition and examples ...
prism where approaches
infinity Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions am ...
. 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 of a bicone as an infinite-sided
bipyramid A (symmetric) -gonal bipyramid or dipyramid is a polyhedron formed by joining an -gonal pyramid and its mirror image base-to-base. An -gonal bipyramid has triangle faces, edges, and vertices. The "-gonal" in the name of a bipyramid does ...
.


See also

*
List of shapes Lists of shapes cover different types of geometric shape and related topics. They include mathematics topics and other lists of shapes, such as shapes used by drawing or teaching tools. Mathematics * List of mathematical shapes * List of two- ...
*
Steinmetz solid In geometry, a Steinmetz solid is the solid body obtained as the intersection of two or three cylinders of equal radius at right angles. Each of the curves of the intersection of two cylinders is an ellipse. The intersection of two cylinders ...
, the intersection of two or three perpendicular cylinders


Notes


References

* * *


External links

*
Surface area of a cylinder
at MATHguide

at MATHguide {{Compact topological surfaces Quadrics Elementary shapes Euclidean solid geometry Surfaces