Cylinder (geometry)

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An elliptic cylinder
An elliptic cylinder

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

\left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\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
A right circular cylinder

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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 = \pi r^2 h \,

and its surface area is:

  • Area of the top is πr2
  • Area of the bottom is πr2
  • Area of the side is 2πrh

Therefore without the top or bottom, the surface area is

A = 2πrh.

With the top and bottom, the surface area is

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

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.)

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:

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

the hyperbolic cylinder:

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

and the parabolic cylinder:

x^2 + 2ay = 0. \,

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