Do Cones and Cylinders Have Faces or Surfaces?

Date: 03/16/2004 at 08:22:25
From: Rhonda
Subject: Does a cone have faces?

Does a cone have faces or are they called curved surfaces and a base?
The same question can be asked for a cylinder.  Are the ends faces or
bases and is the curved surface also considered a face?

I have seen different books with different answers.  Some also say 
that a cone does not have a vertex.

Date: 03/16/2004 at 09:08:30
From: Doctor Peterson
Subject: Re: Does a cone have faces or are they really called curved surf

Hi, Rhonda.

Here are some discussions of this question from our archives:

  Cone, Cylinder Edges?
   http://mathforum.org/library/drmath/sets/select/dm_cone_edge.html 

Too many texts ask this question, which really shouldn't be asked.  
The problem is that there is no standard definition for these terms 
outside of polyhedra.  If they want to talk about faces on a cylinder, 
they need to define the term in a way that applies to it: either
insist that a face must be flat (and, for some, a polygon), and 
provide a different term for curved or round surfaces; or extend the 
definition to cover any distinct surface, flat or curved.  But when 
they do that, they will be disagreeing with someone's usage!  It's 
probably better to use a different term, as some texts do, and just 
call them "surfaces", without talking about "faces" in this context 
at all.

I offer some relevant thoughts on the role of definition at the 
bottom of this long page:

  Definitions as a Tool of Mathematics
    http://mathforum.org/library/drmath/view/62384.html 

Definitions are more flexible than non-mathematicians realize; we take 
a basic idea like "vertex" and adjust it to fit the object we are 
studying (polygons, polyhedra, curves, networks, ...); therefore we 
have to state our definitions at the start of a paper in order to 
clarify how we are using terms.  Elementary texts, on the other hand, 
need to teach standard definitions that are used everywhere, since 
they are only introducing concepts without elaborating them yet, and 
can't just offer a definition for the sake of their discussion. 
Therefore, they need to discuss only those definitions that ARE 
common, and avoid areas like this where no consensus exists--or is 
needed.

Authors and teachers need to ask, what is the value in the concept I 
want to teach?  If it has no value other than introducing the meaning 
of a word, and that word is not clearly defined, then drop it. If it 
is useful (like being able to talk about the parts of a cylinder), 
then choose terms that allow you to do so.  Saying that a cylinder has 
no faces doesn't allow you to talk about the parts of a cylinder, does 
it?  So "face" shouldn't be defined that way for this purpose.  Either 
redefine it, or talk about the "lateral surface" and the "bases" or 
"circular faces" or whatever you want to call them.

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

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