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Polyhedra - Sides? Faces?


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:
The objects you've asked about are solids, not plane figures. The 
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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