Parts of a Cone

Date: 04/18/2001 at 12:59:14
From: Brian McCormick
Subject: Parts of a solid cone

Hello, 

I am a second grade teacher and we are currently teaching a unit on 
shapes. The question came up as to whether or not a solid cone has any 
edges. My contention is that the definition of an edge is where two 
planes intersect, and therefore a cone cannot have an edge. Another 
teacher says that the curved surface of a cone represents an infinite 
number of planes, and therefore represents an infinite number of 
edges. 

I would very much appreciate your response, and don't be afraid to get 
technical. This is as much to satisfy my own curiosity as to let the 
kids know the proper answer.

Brian McCormick

Date: 04/18/2001 at 14:25:21
From: Doctor Peterson
Subject: Re: Parts of a solid cone

Hi, Brian.

We get this question from time to time, and can never really give a 
definite answer. The word "edge" is used in different ways; often 
people get in trouble by introducing the concept of "edge" in the 
context of polyhedra (where it does mean the intersection of two flat 
faces), but then talking about curved surfaces like cones without 
additional comment.

Here's the definition in the Academic Press Dictionary of Science 
and Technology:
 

   1. in graph theory, a member of one of two (usually finite) sets
      of elements that determine a graph; i.e., an element of the edge
      set. The other set is called the vertex set; each element of the
      edge set is determined by a pair of elements of the vertex set...

   2. a straight line that is the intersection of two faces of a
      solid figure.

   3. a boundary of a plane geometric figure.

In the latter sense (which I think is appropriate in discussing a 
cone, even though the dictionary only mentioned plane figures and not 
curved surfaces), the cone has one edge. I definitely would not bring 
in the idea of "an infinite number of edges"; that kind of reasoning 
generally leads to trouble! I would simply say that we can extend the 
concept of edge either from the world of polyhedra (definition 2) or 
from the world of plane geometry (definition 3) to apply to possibly 
curved boundaries of possibly curved surfaces, as long as we say that 
we are doing so. This also agrees with definition 1, which likewise 
does not require straightness (indeed, there is no such concept in 
graph theory), and which relates to boundaries when we consider planar 
graphs (as in Euler's polyhedral formula).

What definition you use depends on what you are going to do with it. 
If you are just describing objects, my loose definition is fine. If 
you are going to prove theorems involving planes and angles, you'll 
want to restrict yourself to the polygonal definition, but then you 
won't be asking any questions about cones. I think people often fail 
to realize that even though we are very particular about definitions 
in math, those definitions may vary from field to field, as they are 
adapted to a certain context. That's what I'm trying to do here.

The same questions arise concerning faces and vertices, and it's even 
harder to decide in those cases.

Here are a couple discussions of related questions from our archives:

   Names of Parts of a Cone
   http://mathforum.org/library/drmath/view/54869.html   

   Types of Cones
   http://mathforum.org/library/drmath/view/55071.html   

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