Definitions of Cones and Cylinders

Date: 03/02/2004 at 12:37:59
From: Ms. Hamilton
Subject: Grade Five Geometry

Are cones and cylinders pyramids, prisms or neither?  My class cannot 
reach an agreement on this subject!

Date: 03/02/2004 at 13:08:45
From: Doctor Peterson
Subject: Re: Grade Five Geometry

Hi, Ms. Hamilton (and class!)

The only way to decide is to look up the definitions and see if they 
fit; the trouble is that you'll find slightly different definitions 
here and there.  And elementary texts aren't known for giving 
mathematically precise, and fully general, definitions, so that might 
not be your best source.

One source of (advanced) definitions is

  Eric Weisstein's World of Mathematics
    http://mathworld.wolfram.com/ 

which is listed in our FAQ.  You might like going through that with 
the class.  As a sample, here are his definitions of "cylinder" and 
"prism":

  In common usage, the term "cylinder" refers to a solid of
  circular cross section in which the centers of the circles all
  lie on a single line (i.e., a right circular cylinder). In
  mathematical usage, "cylinder" is commonly taken to refer to
  only the lateral sides of this solid, excluding the top and
  bottom caps.

  An oblique prism is a polyhedron with two congruent polygonal
  faces and all remaining faces parallelograms (left figure). A
  right prism is a prism in which the top and bottom polygons lie
  on top of each other so that the vertical polygons connecting
  their sides are not only parallelograms, but rectangles (right
  figure). If, in addition, the upper and lower bases are
  rectangles, then the prism is known as a cuboid.

Unfortunately, he doesn't give a good definition of the cylinder in 
the most general sense.  Note that a cylinder in the common, 
specialized sense is really a "right circular cylinder".  It is called 
"circular" because its base is circular; more generally, you can make 
a cylinder from ANY closed plane curve, not just a circle.  And if 
that "curve" is a polygon (to a mathematician, there's nothing wrong 
with a "curve" having straight sides or corners), then you have a 
prism.  So a prism is actually a polygonal cylinder.  Likewise, a 
pyramid is a polygonal cone.  But a (circular) cylinder is not a 
prism, and a (circular) cone is not a pyramid, because the prism and 
the pyramid must be polyhedra, with flat sides.

You can see definitions that fit what I just described in our FAQ:

    http://mathforum.org/dr.math/faq/formulas/faq.cylinder.html 

  A cylinder is a surface generated by a family of all lines
  parallel to a given line (the generatrix) and passing through a
  curve in a plane (the directrix).  A right section is the curve
  formed by the intersection of the surface and a plane
  perpendicular to the generatrix.  The parallel bases of a
  cylinder may form any angle with the axis.

  A cone is a surface generated by a family of all lines through
  a given point (the vertex) and passing through a curve in a
  plane (the directrix).  More commonly, a cone includes the solid
  enclosed by a cone and the plane of the directrix.  The region
  of the plane enclosed by the directrix is called a base of the
  cone.  The perpendicular distance from the vertex to the plane
  of the base is the height of the cone.

To interpret this definition, see

  Definition For Cylinder without Big Words
    http://mathforum.org/library/drmath/view/55052.html 

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/