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Volume of a Regular Octahedron

Date: 09/12/2004 at 10:50:56
From: Jane
Subject: (no subject)

How do you find the volume of a regular octahedron, if you don't know
the height of one of the pyramids?  For example, find the volume of a
regular octahedron with sides of 1 cm.

I know volume = (2/3)AH, but I can't find 'H' and my answer never 
seems right.



Date: 09/12/2004 at 13:02:05
From: Doctor Ian
Subject: Re: (no subject)

Hi Jane,

If it's a _regular_ octahedron, then you can divide it into two 
pyramids, where each pyramid will have all its edge lengths the same.
Does that make sense?

So, you have a regular pyramid with all edges of length L, and you 
want to find the volume.  To use the formula 

      L^2 h
  V = -----
        3

you need to find the height.  To find the height, you'll need to use
the Pythagorean theorem.  To use the Pythagorean theorem, you need to
have a right triangle where you know two of the sides. 

Here is such a triangle:  One vertex, A, is at a base corner of the
pyramid.  A second vertex, B, is at the apex of the pyramid.  The 
third vertex, C, is at the center of the base.

You know that AB is just the length of an edge, which is L.  BC is 
what you're trying to find.  So you need to know AC, the distance from
a corner of the base to the center of the base. 

          L
    A-----------+
    | .         |
    |   .       |
    |     C     |
  L |           |
    |           |
    +-----------+

How do you find this?  You use the Pythagorean theorem again!

As you move on in math, you're going to stop seeing problems where you
can just whip out a formula, fill in the values, and get the answer. 
More and more, you're going to see problems that you'll have to reduce
to smaller and smaller problems, get the answers to the smaller 
problems, and put them back together to get the final answer you're
looking for. 

In this case, the first thing we do is break the problem,

   What is the volume of an octahedron...?

into the smaller problem, 

   The volume of an octahedron is twice the volume of 
   a corresponding pyramid.  

   So what is the volume of one of the pyramids?

For this, we have a formula:

       L^2 h
   V = -----
         3

but to _use_ the formula, we need to know the height.  So we have to
find the height.  If we knew the distance from a corner to the center,
we could find the height.  So now we have to find the distance from
the corner to the center.

And then we can just unwind it all:

   Using the distance from the corner to the center, we 
   can find the height. 

   Using the height, we can find the volume of the pyramid.

   Using the volume of the pyramid, we can find the volume
   of the octahedron.  

In this case, we had a formula that we could use for the volume of the
pyramid.  In many cases, there won't be any such formula.  You'll have
to take a detour and invent one for yourself--inventing the formula 
will be one of your sub-problems, and that will break into 
sub-problems, which will break into more sub-problems, and so on.

This kind of problem reduction, by the way, is the MAIN skill that
you're supposed to be learning in your math classes.  All the earlier
stuff about learning to multiply and divide, and then using variables,
and so on, is just preparation for this kind of problem-solving. 

In other words, it's kind of like:  Okay, now you can stand up, and
walk, and even run a little.  Now it's time to actually learn 
baseball, or soccer, or whatever.  

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Polyhedra

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