Edges, Vertices, Surfaces

Date: 04/22/2003 at 21:37:37
From: Katie
Subject: Edges, vertices, and surfaces

I need help with edges, vertices, and surfaces. How can a shape have 
8 edges and 5 vertices? How can a shape have 12 edges, all the same 
length? How can a shape have 3 surfaces, one curved? 

The one with three surfaces could be a cylinder, but that has 4 
surfaces (I think).

Thanks!

Date: 04/22/2003 at 22:36:44
From: Doctor Peterson
Subject: Re: Edges, vertices, and surfaces

Hi, Katie.

You're right about the cylinder; it is like a soup can, with two flat 
ends and one curved "side."

Probably the biggest difficulty here is to try to visualize the shapes 
and count their edges. I think these problems all relate to familiar, 
ordinary shapes like prisms and pyramids, so if you can find pictures 
of those, you can just count the sides. There are pictures of a lot of 
shapes in the Dr. Math Geometric Formulas FAQ:

   Formulas: Geometric
   http://mathforum.org/dr.math/faq/formulas/ 

Then you need an orderly way to count everything. Let's take a simple 
example, the cube:

       +------+
      /      /|
     /      / |
    +------+  |
    |      |  +
    |      | /
    |      |/
    +------+

You can't see all the edges and surfaces, but you can still think 
about them. First, think about the surfaces. We can group them into 
top, bottom, and sides: there is 1 top, 1 bottom, and 4 sides, making 
a total of 6 surfaces. Now, how about the edges? There are 4 on the 
top, 4 on the bottom, and 4 standing upright between the sides, 
making a total of 12 edges. Hmmm ... they are all the same length, 
too, so you might have an answer here.

Try doing this for other basic shapes, and see if you find one that 
fits the other question. You might be able to see a pattern that lets 
you find the number of parts of any prism or pyramid when you know 
the number of sides in the base.

If you need more help, please write back and show me how far you got.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/