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Q&A #4046


Volume and Surface Area of Cones

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From: Pat Ballew (for Teacher2Teacher Service)
Date: May 30, 2000 at 06:54:04
Subject: Re: Volume and Surface Area of Cones

Well,
  There is no simple explanation for why this is true.  I think I would use
a two fold approach.

First, I would try to find some cylinders and cones with equal radii and
equal heights, and then use rice or water and let the student experiment to
confirm that, at the very least, 1/3 is a good approximation for the
constant.

Then if the student really wants to pursue how mathematicians discovered
this formula, I would suggest walking (very slowly) through the Archimedian
approach using center of gravity.  The concept is not beyond a sixth grader,
they know if from see-saws in the park, but the application of Archimedes is
very deep.   A good example, I believe, can be found in the recent book,
Archimedes, What did he do besides cry Eureka?   This is not an easy thing
for a sixth grader, but it may give them some insight to how difficult some
of the simple formulas were to discover before the creation of Calculus.
(The same formula can be done in calculus as a first semester topic, MUCH
easier, but  a deeper foundation required).

The same 1/3 is true of the volume of ANY prism like volume if one base is
drawn to a point instead of a parallel congruent face.  A square based
pyramid is 1/3 the volume of a square based prism etc.

   I try to tie this together with the two space equivalent that a
triangle (line joined to a point) is 1/2 the area of a line joined to a
congruent line (parallelogram).... and a face in space joined to a point
(pyramid or cone) is 1/3 the volume of a face joined to a face (prism or
cylinder)....  This kind of thing happens again and again in geometry... the
medians of a triangle cut each other in a 2:1 ratio in the plane,,, but in
space the lines joining a vertex of a tetrahedron to the centroid of the
opposite face cut each other in a 3:1 ratio.... To me, it is part of the
beauty of geometry that I come to expect, even count on as part of the
intuitive network I bring to problem solving.....

Hope this helps.  I wish I had an Easy way to make this clear, but I fear,
as always, There is no royal road to geometry...

Cheers,
 -Pat Ballew, for the Teacher2Teacher service

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