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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^2 + 2 \pi 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 cylinder. These are the imaginary elliptic cylinder:

<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>

A cylinder in an engine is the space a piston travels in. The piston is the same size as the two bases of the cylinder (the circular and flat surfaces). In the following drawing, which depicts a cross-section of a steam engine cylinder, the bottom sliding part is the piston, and the top sliding part is a valve that directs steam into the two ends of the cylinder alternately.

The cylinder was also the dominent medium of audio storage from the 1870s to the 1910s, and continued in limited use (such as the dictaphone) through the mid 20th century. See: phonograph cylinder

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