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PostPosted: Wed Oct 21, 2015 12:27 pm 
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Hi
can anybody confirm the number of faces, vertices & Edges?
There are varying answers including in the book if 11+.Thanks


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PostPosted: Wed Oct 21, 2015 12:46 pm 
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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 ...


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PostPosted: Wed Oct 21, 2015 1:05 pm 
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Cone

1 flat Face (the base) which is a circle.
1 curved Face.

1 Vertex (often called the apex)
1 Edge


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PostPosted: Wed Oct 21, 2015 1:06 pm 
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I think this is a very difficult and ambiguous question because it depends on how strictly you define face, vertex and edge.

The instinctive answer would probably be:

1 face (the flat plane section at the bottom)
1 edge (the circular boundary of the face)
1 vertex (the pointy bit at the top)

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.

Alternatively a mathematician might claim that a cone has an infinite number of edges which make up the curved surface but I certainly wouldn't expect a 10 year old to think like that.

I'd be interested to know what the "official" answer is but think it would be wrong if this question was part of 11+ as its so ambiguous IMO.


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PostPosted: Wed Oct 21, 2015 1:14 pm 
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Technically mathematicians would say a face is 2D, otherwise its a surface ... you can't have a curved face.


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PostPosted: Wed Oct 21, 2015 1:22 pm 
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Guest55 wrote:
Technically mathematicians would say a face is 2D, otherwise its a surface ... you can't have a curved face.


What are the technical definitions of a vertex and an edge?


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PostPosted: Wed Oct 21, 2015 2:40 pm 
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In the 11 + book of this website says as edges as 1 but in others as Zero. I also go for Zero by the definition of Edges.

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?


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PostPosted: Wed Oct 21, 2015 3:40 pm 
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An edge is technically where two faces join so a cone has no edges if we say it has one face.


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PostPosted: Wed Oct 21, 2015 4:51 pm 
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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?

Proud_Dad wrote:
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.

Guest55 wrote:
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. :wink:

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. :D

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...

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