When a straight line moves in the space without changing its direction, the ruled surface it sweeps is called a cylindrical surfaceMathworldPlanetmath (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  lm (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 π perpendicularMathworldPlanetmathPlanetmathPlanetmath to any of its rulings, and observe the curve c of intersectionMathworldPlanetmath of π and S, then S is if c is a closed curve.

Figure 1: A closed cylindrical surface

The solid cylindrical surface and two parallel planesMathworldPlanetmath 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 segmentMathworldPlanetmath 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 productPlanetmathPlanetmathPlanetmath of the base area (A) and the height (h):


If the base is a polygonMathworldPlanetmathPlanetmath, the cylinder is called a prism (which is a polyhedron).  The faces of the mantle of a prism are parallelogramsMathworldPlanetmath.  If also the bases of a prism are parallelograms, the prism is a parallelepiped.  If the faces of the mantle of a prism are rectanglesMathworldPlanetmathPlanetmath, one speaks of a right prism, otherwise of a skew prism.

For any integer n3, the following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath statements about a prism P:

  1. 1.

    P has a base that is an n-gon;

  2. 2.

    P has n+2 faces;

  3. 3.

    P has 2n vertices;

  4. 4.

    P has 3n edges.

Note.  The notion of the prism (or cylinder) of a polygon in 3 has a higher-dimensional analogue.  Given any polytope P, the prism of P is the polytope  Prism(P):=P×[0, 1].  The vertices of Prism(P) are the points  (x, 0) and  (x, 1), where x over the vertices of P.  In other words, we drag P a short distanceMathworldPlanetmathPlanetmath through a vector orthogonalMathworldPlanetmathPlanetmath to everything in P, just as we would to obtain the prism of a polygon.

Title cylinder
Canonical name Cylinder
Date of creation 2013-03-22 15:29:21
Last modified on 2013-03-22 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 ParallelotopeMathworldPlanetmath
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