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Given a line *L* and a curve *C* in a plane *P*, the **cylinder**
with **generator** *L* and **directrix** *C* is the surface obtained
by moving *L* parallel to itself, so that a point of *L* is always on
*C*. If *L* is parallel to the *z*-axis, the surface's implicit
equation does not involve the variable *z*. Conversely, any implicit
equation that does not involve one of the variables (or that can be
brought to that form by a change of coordinates) represents a
cylinder.

If *C* is a simple closed curve, we also apply the word
**cylinder** to the solid enclosed by the surface generated in this
way (Figure 1, left).

**Figure 1:** Left: an oblique cylinder with generator *L* and directrix
*C*. Right: a right circular cylinder.

The **volume** contained between *P* and a plane *P'*
parallel to *P* is

where *A* is the area in the plane *P* enclosed by *C*, *h* is the
distance between *P* and *P'* (measured perpendicularly), *l* is the
length of the segment of *L* contained between *P* and *P'*, and
is the angle that *L* makes with *P*. When
=90° we have a **right cylinder**, and *h*=*l*. For a
right cylinder, the **lateral area** between *P* and *P'* is *hs*,
where *s* is the length (circumference) of *C*.

The most important particular case is the **right circular cylinder**
(often simply called a **cylinder**). If *r* is the radius of the
base and *h* is the altitude (Figure 1, right), the
**lateral area** is

2*r*h,

the **total area** is 2*r*(*r*+*h*),
and the **volume** is

*r**h*.

The **implicit equation** of this surface can
be written

*x*+*y*=*r*;

see also Section 16.

*Silvio Levy
Wed Oct 4 16:41:25 PDT 1995*

This document is excerpted from the 30th Edition of the *CRC Standard Mathematical Tables and Formulas* (CRC Press). Unauthorized duplication is forbidden.