Associated Topics || Dr. Math Home || Search Dr. Math

### 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.

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

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search