# Is it correct to state that a cone has no faces?

Faces are attributes of polyhedra, so it doesn't make sense to ask how many faces a cone has.

Are there traditional scholars that use faces attached to cones? How do different countries deal with the subject?

• This is a matter of definitions. If you define a "face" to be a polygon then a circular cone does not have a face. However, if you define a "face" to be a 2-cell in a CW-complex (for example) then you could decompose the cone into a number of faces. I voted to close as "opinion based". Aug 27 at 23:07
• I agree with Steven Gubkin. However, before closing this, it is worth noting that different school textbooks answer this differently, at least in the U.S., so there is no standard answer for educators. I once had an advisory role on a standardized test which included such a question for young children. I pointed out that textbooks disagreed and that the question was not appropriate, but the authors of the test chose to include it anyway since it was in the one textbook they had consulted for this topic. Aug 28 at 0:45
• I would remove "do you agree"s from the question and instead specify the country and level of education; then, answers can tell what is done in that setting and what should be done. Aug 28 at 7:15
• @Scott Eberle: but the authors of the test chose to include it anyway since it was in the one textbook they had consulted for this topic -- Hearing about these types of situations is especially painful to me (and yours is not the only such situation I've heard about), as this has been my primary occupation for 18 years. I can assure you that at least for the tests I deal with, and the people I work with, nothing remotely close to this occurs. See my comments about natural numbers here (continued) Aug 28 at 11:20
• I strongly recommend to read a beautiful book Proofs and Refutations by Imre Lakatos. This question is discussed there in deep details. Aug 31 at 4:51

This isn't a bad question, it's a good opportunity to discuss how mathematical definitions are made and their relative merits.

In convex geometry, a face of a convex body $$K$$ is the intersection $$K \cap H$$ where $$H$$ is a supporting hyperplane of $$K$$. (See, for example, here.) So a cone has infinitely many faces -- the base disc, and also the infinitely many line segments joining the cone point to each of the points on the perimeter of the base disc.

I'm sure there are different definitions in other fields.

This answer explains why this question can't be answered (but it does not fit inside a comment section).

First of all: what is a cone?
According to this Wikipedia-page, a cone has a circular ground surface, which is in most mathematics handbooks pure nonsense:

• Either a cone does not have a ground surface.
• Either the ground surface is a general ellips (not necessary a circle).

Please clearly define what you mean by a cone.

Next: what is a face? This Wikipedia-page mentions different definitions of that word, but the geometrical definition is not present.
So, a face: is this a subset of a plane, where every two points can be connected by a line, never leaving the face? Or do you have a better definition?