In mathematics, a cylinder is a quadric, i.e. a three-dimensional 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.

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 h, then its volume is given by

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

and its 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.

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:

$x^2\; +\; 2y\; =\; 0\; \backslash ,$

External links

• Spinning Cylinder Math Is Fun

Elementary geometry | Euclidean solid geometry

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