cylinder
When a straight line moves in the space without changing its direction, the ruled surface it sweeps is called a cylindrical surface^{} (or, in some special cases, simply a cylinder). Formally, a cylindrical surface $S$ is a ruled surface with the given condition:
If $p,q$ are two distinct points in $S$, and $l$ and $m$ are the rulings passing through $p$ and $q$ respectively, then $l\parallel m$ (this includes the case when $l=m$).
If the moving line returns to its starting point, the cylindrical surface $S$ is said to be . In other words, if we take any plane $\pi $ perpendicular^{} to any of its rulings, and observe the curve $c$ of intersection^{} of $\pi $ and $S$, then $S$ is if $c$ is a closed curve.
The solid cylindrical surface and two parallel planes^{} is a cylinder. The portion of the surface of the cylinder belonging to the cylindrical surface is called the lateral surface or the mantle of the cylinder and the portions belonging to the planes are the bases of the cylinder.
The bases of any cylinder are congruent. The line segment^{} of a generatrix between the planes is a of the cylinder. All side lines are equally long. If the side lines are perpendicular to the planes of the bases, one speaks of a right cylinder, otherwise of a skew cylinder.
The perpendicular distance of the planes of the bases is the of the cylinder. The volume ($V$) of the cylinder equals the product^{} of the base area ($A$) and the height ($h$):
$$V=Ah$$ 
If the base is a polygon^{}, the cylinder is called a prism (which is a polyhedron). The faces of the mantle of a prism are parallelograms^{}. If also the bases of a prism are parallelograms, the prism is a parallelepiped. If the faces of the mantle of a prism are rectangles^{}, one speaks of a right prism, otherwise of a skew prism.
For any integer $n\ge 3$, the following are equivalent^{} statements about a prism $P$:

1.
$P$ has a base that is an $n$gon;

2.
$P$ has $n+2$ faces;

3.
$P$ has $2n$ vertices;

4.
$P$ has $3n$ edges.
Note. The notion of the prism (or cylinder) of a polygon in ${\mathbb{R}}^{3}$ has a higherdimensional analogue. Given any polytope $P$, the prism of P is the polytope $\text{Prism}(P):=P\times [0,\mathrm{\hspace{0.17em}1}]$. The vertices of $\text{Prism}(P)$ are the points $(x,\mathrm{\hspace{0.17em}0})$ and $(x,\mathrm{\hspace{0.17em}1})$, where $x$ over the vertices of $P$. In other words, we drag $P$ a short distance^{} through a vector orthogonal^{} to everything in $P$, just as we would to obtain the prism of a polygon.
Title  cylinder 
Canonical name  Cylinder 
Date of creation  20130322 15:29:21 
Last modified on  20130322 15:29:21 
Owner  stevecheng (10074) 
Last modified by  stevecheng (10074) 
Numerical id  12 
Author  stevecheng (10074) 
Entry type  Topic 
Classification  msc 51M20 
Classification  msc 51M04 
Related topic  Parallelotope^{} 
Defines  cylindrical surface 
Defines  lateral surface 
Defines  mantle 
Defines  base 
Defines  side line 
Defines  right cylinder 
Defines  skew cylinder 
Defines  prism 
Defines  parallelepiped 
Defines  right prism 
Defines  skew prism 