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Q&A #4046


Volume and Surface Area of Cones

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From: Jeanne (for Teacher2Teacher Service)
Date: May 30, 2000 at 15:08:32
Subject: Re: Volume and Surface Area of Cones

I wish more of my students' parents feel the same as you!

I agree with Pat, there is no easy way to explain the "1/3" part of the
formula.  The best I can do is offer as "convincing argument" as I can.  For
some the "pouring sand from a pyramid to a cube  3 times"  is convincing
enough.  For others, I offer the following:

First, let's talk about "conservation of volumes" using a deck of cards or a
ream of paper.  When I talk to my students, I have them compute the volume
of the deck/ream stacked so that the cards/paper form a rectangular prism
with all of the side perpendicular to the base.

Then, we discuss what happens to the volume of the prism when we make
a "crooked" (skewed) prism.  That is, we slide the deck/ream so that sides
no longer form right angles with the base.  The kids usually see that the
volume remains the same  because we haven't removed anything.  We've just
slid them around.

Besides the volume, what remains the same between the original prism and
the "crooked" prism are the bases and the heights.

On to the pyramid...
Given a pyramid and a "crooked" pyramid.  As long as the bases are congruent
and the heights are the same lengths,  the volumes of the two pyramids are
equal.  I have models of pyramids made from stacks of smaller and smaller
cardboard squares that I move around the same way I do a deck of cards.


Now for the 1/3 part of the formula...Remember I am hoping to provide you
with something more convincing than "3 fill-ups"...

You can create a 3 copies of a square based pyramid which will fit together
to form a cube.        Please excuse my "graphics."  I am really limited by
                *X    the word processor here.
               **          The template here is supposed to be a square with
             *  *      2 right triangles attached to the square as shown.
           *    *           The triangles have legs which are the same
          ************Y       length as the sides of the square.
          *     *   *     Fold along the sides of the square so that the
          *     *  *     corner points X and Y meet.  Tape to secure.  This
          *     * *     should look like the corner of a room.  If you take
         Z*******       a piece of spaghetti so that one end is on point Z
and the other touches points X and Y, you'll complete the pyramid.

Make the other 2 pyramids.

Now you have a puzzle.  These three pyramids can be fitted together to form
a cube.  Hence, one pyramid has the volume that is 1/3 of the volume of the
cube.

Remember these 3 pyramids have the same volume as other pyramids with bases
congruent to these squares and whose heights are the same as these models.

This IS NOT a proof of the "1/3" but perhaps it is more of something you
were looking for.

If you'd like to take it further, these 3 pyramids have the same volume as
other pyramids whose bases have the same area as the square and whose
heights are the same as these models.

Hope this helps.

-Jeanne, for the Teacher2Teacher service

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