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Math Geek
 
 
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Math Geek
 
 
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Old 05-12-2014, 10:17 PM
 
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Personally, I believe the definition of a cone evolves with school level, just as other math concepts evolve with grade level and deepen as their developing minds can comprehend them.

At the elementary level, students are going to learn the basics about 3-D figures from a 2-D perspective, as young minds cannot process the complexities of shapes from a 3-D perspective. Textbooks are not going to get into the difference of a face and a base and the details of all the definitions behind 3-D shapes. But at the elementary level they should be calling the flat surface a base, not a face. From a 2-D perspective, it looks like 2 lines that meet at a point on a cone, which by definition, is a vertex. So that may be the reason why the point of a cone is a called a vertex sometimes at the elementary level.

As students progress through school, student's are taught about nets in order to help bridge their minds from the 2-D world to the 3-D world. The net of a cone is actually a circle and a sector of a circle, both of which are not polygons, therefore still would not have any faces.

By the time they reach High School, students' minds are capable of thinking in 3-D. They learn the complexities of 3-D shapes from a 3-D perspective, but in a finite context. In a finite context, a face of a polyhedron is any of the polygons that make up it's boundaries. A cone's base is a circle, not a polygon, therefore not a face. Also in this context, a vertex is defined as a point where multiple faces meet. Therefore, with no faces, a cone cannot have a vertex. Furthermore, an edge is a straight line where two faces meet- no faces, no edges. So the definition of a cone from a finite perspective has a base, no faces, no edges, and no vertex. Also students typically work with right circular cones, and oblique cones are introduced later on.

When students reach calculus, they learn the concepts of limits and infinity. A cone's definition then will change in the infinite context as others have mentioned. The limit of an n-gon as n reaches infinity is, by defintion, a circle. So people can argue that a circle is indeed a polygon and then a cone's base is also a face. Also, it's rounded surface is comprised of an infinite number of triangles extending from the axis of a cone. Since triangles are polygons, one can argue that a cone has a vertex because it is a point where an infinite number of polygons meet.

So I believe the confusion lies in the fact that people do not clearly state where they are in the evolution of a cone when stating their defintion. It creates a lot of confusion when definitions are being thrown out there on the internet without specifying their target audience. But it is clearly a great example of having student's think more deeply into mathematics and expand the possibilities! It becomes a great classroom discussion!
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