Parts of a Cone
Date: 04/18/2001 at 12:59:14 From: Brian McCormick Subject: Parts of a solid cone Hello, I am a second grade teacher and we are currently teaching a unit on shapes. The question came up as to whether or not a solid cone has any edges. My contention is that the definition of an edge is where two planes intersect, and therefore a cone cannot have an edge. Another teacher says that the curved surface of a cone represents an infinite number of planes, and therefore represents an infinite number of edges. I would very much appreciate your response, and don't be afraid to get technical. This is as much to satisfy my own curiosity as to let the kids know the proper answer. Brian McCormick
Date: 04/18/2001 at 14:25:21 From: Doctor Peterson Subject: Re: Parts of a solid cone Hi, Brian. We get this question from time to time, and can never really give a definite answer. The word "edge" is used in different ways; often people get in trouble by introducing the concept of "edge" in the context of polyhedra (where it does mean the intersection of two flat faces), but then talking about curved surfaces like cones without additional comment. Here's the definition in the Academic Press Dictionary of Science and Technology: 1. in graph theory, a member of one of two (usually finite) sets of elements that determine a graph; i.e., an element of the edge set. The other set is called the vertex set; each element of the edge set is determined by a pair of elements of the vertex set... 2. a straight line that is the intersection of two faces of a solid figure. 3. a boundary of a plane geometric figure. In the latter sense (which I think is appropriate in discussing a cone, even though the dictionary only mentioned plane figures and not curved surfaces), the cone has one edge. I definitely would not bring in the idea of "an infinite number of edges"; that kind of reasoning generally leads to trouble! I would simply say that we can extend the concept of edge either from the world of polyhedra (definition 2) or from the world of plane geometry (definition 3) to apply to possibly curved boundaries of possibly curved surfaces, as long as we say that we are doing so. This also agrees with definition 1, which likewise does not require straightness (indeed, there is no such concept in graph theory), and which relates to boundaries when we consider planar graphs (as in Euler's polyhedral formula). What definition you use depends on what you are going to do with it. If you are just describing objects, my loose definition is fine. If you are going to prove theorems involving planes and angles, you'll want to restrict yourself to the polygonal definition, but then you won't be asking any questions about cones. I think people often fail to realize that even though we are very particular about definitions in math, those definitions may vary from field to field, as they are adapted to a certain context. That's what I'm trying to do here. The same questions arise concerning faces and vertices, and it's even harder to decide in those cases. Here are a couple discussions of related questions from our archives: Names of Parts of a Cone http://mathforum.org/library/drmath/view/54869.html Types of Cones http://mathforum.org/library/drmath/view/55071.html - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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