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Q&A #4046 |
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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