Volume of a PyramidDate: 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/ |
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