caring star wrote:
But just wondering which answer to follow for 11+ exam, as in the exam: they just ask How many edges for a cone? What is the answer - 0 or 1?
But then an edge is often defined as the line where 2 faces meet. Therefore because there is only 1 face there can be no edges. Similarly a vertex is defined as the point where multiple edges meet so you could say it has no vertices.
The reason you will get different answers is the way some people define 'face'. Some think a cone has one face (a 2D circle), no vertices and no edges. Others include curved surfaces as faces ...
I agree with Proud_Dad that this question is too ambiguous to be used in an 11+ paper.
As Guest55 suggested, it all depends on a definition of a face.
According to 'Oxford Study Mathematics Dictionary':
- a face is a plane (flat) surface enclosed by an edge or edges;
- an edge (in a 3D shape) is defined as a straight line where two faces meet;
- a vertex is an angular point where 3 or more edges meet.
These definitions apply to polyhedra, i.e. 3D shapes whose faces are all polygons. A polygon is a flat shape completely enclosed by 3 or more straight edges.
Based on the above definitions, a cone is not a polyhedron because its base (circle) is not a polygon and its side surface is curved, not flat, so it is not considered a face.
The same dictionary states that a 'vertex of a cone is the fixed point used in making it', in other words the cone's apex. Unfortunately, the same dictionary doesn't go into detail of whether a cone has an edge or not, but at least we know it has a vertex/apex.
If a cone is not a polyhedron, then face, edge and vertex definitions that apply to polyhedra do not necessarily apply to a cone.
If a face has to be flat and an edge is where two faces meet, a cone can't have an edge because it does not have two flat faces that would meet making a straight line edge.
If, however, we assume that a curved surface of the cone can be considered a face, the cone would have two faces by using such a definition and therefore it would have an edge.
Apart from deciding whether the curved surface of a cone can be considered a face or not, it is about being consistent. It is either 2 faces and 1 edge, or 1 face and 0 edges.
Personally, I would go with one face (the circle), one curved surface, one edge and one vertex.
The ultimate 'get-out' clause is that the standard polyhedron definitions do not apply here, because a cone is not a polyhedron.
There are a lot of places with discussions about this topic, one of them here:http://mathforum.org/library/drmath/view/54681.html
Having said all that - I am not a mathematician, so please correct me if I am wrong.
Edit: PS. Apologies for a long-winded post, but I was somewhat confused by this cone dilemma, so wanted to get to the bottom of it...