In mathematics a cylinder is a quadric, i.e. a three-dimensional surface, with the following equation in Cartesian coordinates:

<math>\left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = 1<math>

This equation is for an elliptic cylinder. If a = b then the surface is a circular cylinder. 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

<math>V = \pi r^2 h<math>

and its surface area is

<math>A = 2 \pi r ( r + h )<math>

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:

<math>\left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = -1<math>

the hyperbolic cylinder:

<math>\left(\frac{x}{a}\right)^2 - \left(\frac{y}{b}\right)^2 = 1<math>

and the parabolic cylinder:

<math>x^2 + 2y = 0<math>

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