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### Definitions of Cones and Cylinders

```Date: 03/02/2004 at 12:37:59
From: Ms. Hamilton

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

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

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/
```
Associated Topics:
Elementary Definitions
Elementary Polyhedra
Elementary Three-Dimensional Geometry
High School Definitions
High School Higher-Dimensional Geometry
High School Polyhedra
Middle School Definitions
Middle School Higher-Dimensional Geometry
Middle School Polyhedra

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