Hi Merium,

We have a number of responses for you.

This is a 'typical' question where the teacher is trying to extend the applications of Euler's formula, where we do not usually have the requirements of a connected planar graph!

I think I have posted some previous answers to this (here and elsewhere).

There is a context where it makes sense - building a closed cone with pieces of paper and tape.
- there are two pieces of paper (faces)
- there are two pieces of tape used (2 edges) and
- there are two places where the pieces of tape end (vertices)
2 - 2 + 2 = 2.

Walter

According to wikipedia, a cone "is the solid figure formed by the locus of all straight line segments that join the apex to the base." Thus it is a three dimensional object.

If the base of the cone is a polygon with n edges then the cone is called a pyramid. Each edge on the perimeter of the base, when joined with the apex of the pyramid, will form a triangle which has 3 sides. For each edge of the base there is one such triangular face of the cone. Thus the edges of a pyramid are the n edges of the base plus n additional edges which join the vertices of the base to the apex.

If the base of the cone is a smooth curve, e.g. a circle then we may want to re-think what the word "edge" means in this case since a simple straight line segment does not seem to fit. Think about a round table for
example, we all know what it means for something to fall off the edge of the table. In this case the "edge" is simply the perimeter of the base, and the sides of the cone have no edges.

Use these ideas to explore the notion of "edge" for a cone that has a base which has some curvy parts and some straight parts. What do you think?

Dr. L. Dame

I suggest that you look a previous responses we have given to this and similar questions, In particular

mathcentral.uregina.ca/QQ/database/QQ.09.08/h/deb1.html

mathcentral.uregina.ca/QQ/database/QQ.09.06/h/erica1.html

and

mathcentral.uregina.ca/QQ/database/QQ.09.06/s/felicia1.html

Harley