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I'm interested to know if anyone can point me to a non-calculus way of seeing that the $\text{volume of a pyramid} = \frac{1}{3}\times(\text{area of base})\times(\text{height})$. Yes, I've googled.

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  • $\begingroup$ There are many kinds of pyramids. Do you have any particular type in mind? For example, one might take a convex n-gon and take the convex hull with a point that does not lie in the plane of the n-gon. $\endgroup$ – Joseph Malkevitch Nov 4 '10 at 21:51
  • $\begingroup$ @Joseph: He meant the one with a square base I suppose. $\endgroup$ – J. M. is a poor mathematician Nov 4 '10 at 22:58
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I actually find it easiest to take a cube and divide it into 6 pyramids: just connect all of the corners of a cube to the midpoint, and for each of the 6 faces of the cube, you get a pyramid having that face as its base. If the cube has a side length s, then each pyramid has an s by s base and height s/2, and its volume is 1/6 the volume of the cube, which is 1/3 base times height.

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The easiest intuitive one I know is to take a cube and divide it into three pyramids. Take one corner and form the pyramid with each of the three faces that don't touch that corner. They are congruent, so have volume 1/3*base*height as the base is a square the side of the cube and the height is the side of the cube as well. Then appeal to affine transformations to say this applies to all pyramids.

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    $\begingroup$ There is a visual demonstration of this in this SE answer: math.stackexchange.com/questions/623/… $\endgroup$ – Derek Jennings Nov 4 '10 at 14:35
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    $\begingroup$ I needed to visualize this so I found a template for a paper model cut it out and now I see it! Thanks. www.ms.uky.edu/algebracubed/lessons/Dissecting_the_Cube_lesson.pdf $\endgroup$ – Sol Lederman Nov 4 '10 at 15:08
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Proof:

Put together 6 pyramids to make a cube. You will see that volume of whole cube can be considered as sum of 2 equal smaller volumes (area by half-height of cube). Its also the height of pyramid multiplieds by area. The small volume is also the sum of 3 pyramids, because 3 is half of 6. So single pyramid is 1/3rd of this multiplication.

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    $\begingroup$ How is this different from Gabe's answer? $\endgroup$ – J. M. is a poor mathematician Nov 5 '10 at 0:23
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    $\begingroup$ oh, sorry, did not mean to duplicate it. I did not see someone beat me on it $\endgroup$ – user3140 Nov 5 '10 at 1:18

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