Cylinder_(geometry)

An elliptic cylinder

In mathematics, a cylinder is a quadric surface, with the following equation in Cartesian coordinates:

$\backslash left(\backslash frac\{x\}\{a\}\backslash right)^2\; +\; \backslash left(\backslash frac\{y\}\{b\}\backslash right)^2\; =\; 1.$

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 can be any curve.

The cylinder is a degenerate quadric because at least one of the coordinates (in this case z) does not appear in the equation. By some definitions the cylinder is not considered to be a quadric at all.

A right circular cylinder

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 r and length (height) h, then its volume is given by

$V\; =\; \backslash pi\; r^2\; h\; \backslash ,$

and its surface area is:

• Area of the top is $\backslash pi\; r^2$
• Area of the bottom is $\backslash pi\; r^2$
• Area of the side is $2\; \backslash pi\; r\; h$

Therefore without the top or bottom, the surface area is

$A\; =\; 2\; \backslash pi\; r\; h.$

With the top and bottom, the surface area is

$A\; =\; 2\; \backslash pi\; r^2\; +\; 2\; \backslash pi\; r\; h\; =\; 2\; \backslash pi\; r\; (\; r\; +\; h\; ).\backslash ,$

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.)

Other types of cylinders

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:

$\backslash left(\backslash frac\{x\}\{a\}\backslash right)^2\; +\; \backslash left(\backslash frac\{y\}\{b\}\backslash right)^2\; =\; -1$

the hyperbolic cylinder:

$\backslash left(\backslash frac\{x\}\{a\}\backslash right)^2\; -\; \backslash left(\backslash frac\{y\}\{b\}\backslash right)^2\; =\; 1$

and the parabolic cylinder:

$x^2\; +\; 2ay\; =\; 0.\; \backslash ,$

See also

• Steinmetz solid, the intersection of two or three perpendicular cylinders
• Prism (geometry)

External links

Wikimedia Commons has media related to:

Cylinder (geometry)

• Surface Area MATHguide
• Volume MATHguide
• Spinning Cylinder Math Is Fun
• calculate surface area and volume with your own values
• Paper model cylinder
• Volume of a cylinder Interactive animation

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