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### Polyhedra - Sides? Faces?

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Date: 10/13/97 at 19:13:19
From: Maria Carrow
Subject: Polyhedra

How many sides does a tetrahedron have?
How many sides does an icosahedron have?
```

```
Date: 10/14/97 at 16:23:04
From: Doctor Chita
Subject: Re: Polyhedra

Dear Maria:
ending "-hedron" tells you that these are three-dimensional shapes
that have "faces" rather than "sides" like a triangle or a square.

A regular "polyhedron" is a solid having faces (surfaces) in the shape
of a regular polygon. The other parts of a polyhedron are called
edges, where the faces meet, and vertices, corners where vertices of
the faces coincide. [What would hurt if you sat on one. :-) ]

The prefix of each word tells you something about the solid. In the
case of a tetrahedron,  the prefix "tetra-" means four. Therefore,
there are four triangular faces in a tetrahedron.

An icosahedron has 20 triangular faces. The prefix "icosa-" is from
the Greek word "eikosi" meaning 20.

These two regular polyhedra (plural of "polyhedron") are special
because they are two of the five Platonic solids, the simplest of
which is the tetrahedron. The others are the cube (with 6 square
faces, as you know), the octahedron (with 8 triangular faces), and the
dodecahedron (with 12 faces that are regular pentagons).

There is also an interesting relationship among the number of faces
(f), edges (e), and vertices (v) of a polyhedron. The mathematician
Leonard Euler discovered that in every polyhedron, not just the
Platonic solids,  f - e + v = 2.

For example, in a cube, f = 6, e = 12, and v = 8, and 6 - 12 + 8 = 2.
Try this relation on other polyhedra - for example, the Great Pyramid
in Egypt. (Remember, it has a rectangular base.)

I'll bet you didn't think the answer to your question would be so
interesting, did you? Hope this helps.

-Doctor Chita,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Geometry
Middle School Polyhedra

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