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Cylinder (geometry)
 
 
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A cylinder is one of the most curvilinear basic geometric shapes: the surface
Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space E3....
 formed by the points at a fixed distance from a given straight line, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder.






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Cylinder Geometry
A cylinder is one of the most curvilinear basic geometric shapes: the surface
Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space E3....
 formed by the points at a fixed distance from a given straight line, 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
Surface area

Surface area is how much exposed area an object has. It is expressed in square units. If an object has flat Face , its surface area can be calculated by adding together the areas of its faces....
 and the volume
Volume

The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimensional space it occupies, often quantified numerically....
 of a cylinder have been known since deep antiquity.

In differential geometry, a cylinder is defined more broadly as any ruled surface
Ruled surface

In geometry, a surface is ruled if through every point of there is a straight line that lies on . The most familiar examples are the plane and the curved surface of a cylinder or cone ....
 spanned by a one-parameter family of parallel lines. A cylinder whose cross section
Cross section

Cross section can refer to:* Cross section , a three-piece rock band from the Mornington Peninsula, Australia* Cross section , the intersection of a 3-dimensional body with a plane...
 is an ellipse
Ellipse

In mathematics, an ellipse is the apparent shape of a circle viewed obliquely from outside it, as distinct from a hyperbola which is the shape seen from inside....
, parabola
Parabola

In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface....
, or hyperbola
Hyperbola

In mathematics a hyperbola is a smooth function planar curve having two connected components or branches, each a mirror image of the other and resembling two infinite bow aimed at each other....
  is called an elliptic cylinder, parabolic cylinder, or hyperbolic cylinder. A prism
Prism (geometry)

In geometry, an n-sided prism is a polyhedron made of an n-sided polygon base, a Translation copy, and n faces joining corresponding sides....
 is a cylinder whose cross-section is a polygon.

Common usage


In common usage, a cylinder is taken to mean a finite section of a right circular cylinder with its ends closed to form two circular surfaces, as in the figure (right). If the cylinder has a radius
RADIUS

Remote Authentication Dial In User Service is a networking protocol that provides centralized access, authorization and accounting management for people or computers to connect and use a network service....
 r and length (height) h, then its volume
Volume

The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimensional space it occupies, often quantified numerically....
 is given by

and its surface area
Surface area

Surface area is how much exposed area an object has. It is expressed in square units. If an object has flat Face , its surface area can be calculated by adding together the areas of its faces....
 is:

  • the area of the top +
  • the area of the bottom +
  • the area of the side .


Therefore without the top or bottom (lateral area), the surface area is

With the top and bottom, the surface area is

For a given volume, the cylinder with the smallest surface area has h = 2r. For a given surface area, the cylinder with the largest volume has h = 2r, i.e. the cylinder fits in a cube (height = diameter.)

Cylindric section

Cylindric sections are the intersections of cylinders with planes. Although these mostly yield ellipses (or circles), a degenerate case of two parallel lines, known as a ribbon, can also be produced, and it is also possible for there to be no intersection at all.

Other types of cylinders

Elliptic Cylinder
An elliptic cylinder is a quadric surface, with the following equation in Cartesian coordinates:

This equation is for an elliptic cylinder, a generalization of the ordinary, circular cylinder (a = b). Even more general is the generalized cylinder: the cross-section
Cross section (geometry)

In geometry, a cross-section is the intersection of a body in 2-dimensional space with a line, or of a body in 3-dimensional space with a plane, etc....
 can be any curve.

The cylinder is a degenerate quadric
Quadric

In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface defined as the locus of root of a quadratic polynomial....
 because at least one of the coordinates (in this case z) does not appear in the equation.

An oblique cylinder has the top and bottom surfaces displaced from one another.

There are other more unusual types of cylinders. These are the imaginary elliptic cylinders:

the hyperbolic cylinder:

and the parabolic cylinder:

See also

  • Steinmetz solid
    Steinmetz solid

    In geometry, the Steinmetz solid is the solid body generated by the intersection of two or three cylinder of equal radius at right angles. It is named after Charles Proteus Steinmetz, though these solids were known long before Steinmetz studied them....
    , the intersection of two or three perpendicular cylinders
  • Prism (geometry)
    Prism (geometry)

    In geometry, an n-sided prism is a polyhedron made of an n-sided polygon base, a Translation copy, and n faces joining corresponding sides....


External links

  • at MATHguide
  • at MATHguide
  • at Math Is Fun
  • Interactive animation at Math Open Reference