Volume of a Pyramid

Date: 05/29/2002 at 05:34:52
From: Michelle
Subject: pyramids

Why does the formula of the volume of pyramid have a 1/3 in it?

Date: 05/29/2002 at 09:23:05
From: Doctor Rick
Subject: Re: pyramids

Hi, Michelle.

There is an extended discussion of several ways to prove the formula 
for the volume of a pyramid in the Dr. Math Archives:

  Volume of a Pyramid
  http://mathforum.org/library/drmath/view/55041.html

Here is a partial answer to your question about 1/3; not a complete 
answer but something to get you thinking. 

I can easily show that the volume of one particular pyramid 
(admittedly a little odd-looking) is 1/3 the base area times the 
altitude. This pyramid has a square base and a vertical edge coming up 
from one vertex of the square, the same height as the edges of the 
square. Join the top of this vertical edge to each of the other three 
vertices of the square, and you have my pyramid. 

In this picture I'm showing all the edges, including the edges that 
are hidden on the back of the pyramid -- as if it were transparent.

   +
   |\\ \
   | \ \  \
   |  \   \  \
   |    \   \   \
   |     \     \   \
   |      \      \    \
   |       \        \    \
   |         \        +     \
   |          \  /         \   \
   |        /  \                \ \
   |   /        \                   \\
   +              \                     +
       \           \                /
            \       \           /
                 \   \     /
                      +

You can make such a pyramid out of cardboard or index cards -- it's a 
nice project.  The hardest part is to find the shape of the two 
diagonal sides. Make three of the pyramids. 

You'll find (maybe you can visualize this without actually making the 
pyramids) that the three pyramids can be put together to make one 
whole cube! Put the three "pointy ends" together, with the square 
bases of the other two pyramids vertical, one on the front right side 
and one on the back right side of the figure above.

Do you get it now? The volume of the cube is s*s*s if the sides are 
of length s. The volume of each of the identical pyramids must 
therefore be (1/3)s*s*s. The height (h) is s, and the area of the 
base (b) is s*s, so the area of this pyramid is (1/3)b*h.

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