Does a Cone have an Edge? A Vertex?

Date: 03/12/2002 at 16:07:05
From: Mark Homan
Subject: 4th grade geometry, cone questions

Dear Dr. Math, 

Our 4th grade math textbook defines a cone as "A solid figure with 
one circular face and one vertex."  This sounds reasonable until you 
read the textbook's definitions for face, edge, and vertex. The 
textbook defines a face as "A flat surface of a solid."  It defines an 
edge as "A line segment where two faces of a solid meet." It defines a 
vertex as "A point where two or more edges meet."  

Assuming that these definitions are accurate and that I'm not 
misinterpreting them, a cone must not have a vertex. If a cone has 
only one face, then it can't possibly have an edge. Therefore, if it 
doesn't have an edge, it can't have a vertex.  

Please help clear this up.
Thank you in advance.
Mark Homan

Date: 03/12/2002 at 16:34:16
From: Doctor Peterson
Subject: Re: 4th grade geometry, cone questions

Hi, Mark.

This is an old favorite question of mine! Here is a discussion of this 
issue from our archive, with links to more of the same:

   Parts of a Cone
   http://mathforum.org/library/drmath/view/55301.html   

Elementary texts (and high school texts, for that matter) are not 
always very careful about definitions. The problem really is that the 
same word can be used with slightly different but related definitions, 
and we don't always bother to specify how to modify the definitions 
when we move to a different context. The definitions given are for a 
polyhedron. When you talk about a cone or cylinder, you have to either 
use a different set of words, since "edge" and "vertex" as defined 
don't apply at all, and "face" applies only to one of the two surfaces 
of a cone; or you have to modify the definitions to allow curved edges 
and faces. Taking the latter approach, the cone will have two faces, 
one curved, and one curved edge. I'm not sure I've ever seen such 
modified definitions actually stated, but I have no trouble allowing 
them, as long as we state them clearly! What really bothers me is when 
a book is consistent enough not to call the curved "face" of a cone a 
face, but doesn't bother to define a word that we _can_ use for it. 
When they then go ahead and say it does have an edge or a vertex, 
children are bound to be confused.

The really tricky part here is that the "vertex" of a cone has nothing 
to do with edges, so it needs a whole new definition; and I can't 
think of a really good elementary-level definition for what they 
obviously mean, which is simply a "point." I prefer to use the word 
"apex" and avoid the problem.

By the way, I hope what you quoted is not given as the _definition_ of 
a cone, but just a _description_. It certainly doesn't distinguish a 
cone from other shapes I can imagine. We define it in our FAQ, but it 
can be hard to say it in a way young children can grasp easily.

   Cone Formulas - Dr. Math FAQ
   http://mathforum.org/dr.math/faq/formulas/faq.cone.html   

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