Pyramids and Triangular Prisms

Date: 05/09/2000 at 16:14:56
From: Victoria
Subject: Math Help

Dear Dr. Math,

I need help finding shapes of triangular prisms. I get pyramids and 
triangular prisms mixed up. What's the difference between them?

Date: 05/09/2000 at 21:09:34
From: Doctor Peterson
Subject: Re: Math Help

Hi, Victoria.

One of the things that make math more pleasant is seeing what the 
words are all about. So let's look at what a prism and a pyramid are.

In either case, you start with some plane figure - a triangle, a 
square, or whatever. To make a prism, just make a second copy and move 
it somewhere outside the plane, keeping it parallel - that is, just 
slide it straight up. (If you slide it vertically, so it's exactly 
over the base, we call it a "right prism," meaning it goes up at right 
angles to the base. If you slide it up at an angle, it's an "oblique 
prism.") 

Say you've drawn a triangle on your desk. Put an identical triangle on 
top of it and lift the second triangle straight up above the desk. Now 
imagine connecting each vertex of the bottom triangle to the 
corresponding vertex of the top triangle. You've just made a right 
triangular prism:

                 +
             /   :\
         /       : \
     +--------------+
     |           :  |
     |           +  |
     |       .    . |
     |   .         .|
     +--------------+

The volume of a prism is just the product of the area of the base and 
its height; think of it as a stack of pancakes whose volume is the 
product of the size of each pancake and the number of pancakes.

To make a pyramid, you start with a base as before. But this time, 
instead of putting a copy of the base above it, just make a single 
point somewhere up in the air, and connect every vertex of the base to 
that one point:

                 +
               / |\
             /   |\
           /     + \
         /   .    .\
       / .         .\
     +--------------+

The volume of a pyramid is obviously less than a prism of the same 
height; it's a stack of pancakes that get smaller and smaller as you 
go up. It turns out that it's exactly 1/3 the volume of the prism.

For pictures of prisms and pyramids, see the Dr. Math FAQ:

  Prism Formulas
  http://mathforum.org/dr.math/faq/formulas/faq.prism.html   

  Pyramid and Frustum Formulas
  http://mathforum.org/dr.math/faq/formulas/faq.pyramid.html   

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