cars - Cylinder (geometry)

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

V = π r 2 h

and its surface area is

A = 2π 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.

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 + 2 y = 0

Categories: Elementary geometry