



 
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. 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 rethink 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 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 http://mathcentral.uregina.ca/QQ/database/QQ.09.08/h/deb1.html http://mathcentral.uregina.ca/QQ/database/QQ.09.06/h/erica1.html and http://mathcentral.uregina.ca/QQ/database/QQ.09.06/s/felicia1.html Harley  


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