My textbook says that the ancient Egyptians found the formula for the base of a pyramid that has the top cut off, which is 1/3 H (B^{2}+ bB + b^{2}). H is the height, B is the side of the bottom base, and b is the side of the top base. It is a mystery how the Egyptians found this since they didn't know algebra. The only surviving example shows how to find the volume of the bottom part of a pyramid that has been cut in half. I don't like mysteries so I tried to imagine what could have happened. The result seems correct, but way too simple. Please see if I am overlooking something or if it is a plausible explanation.
I know when I cut a pyramid out of a block of clay, the volume of that pyramid is 1/3 of the block. More of that volume is in the bottom half. I wonder: what could be the volume of the bottom half of the pyramid? Let’s start with a fresh block of clay. Hmm, I could cut out a pyramid and then cut it in half, but that’s a lot of work and it would be hard to find the volume of the bottom piece when I’m done. Why don’t I just leave the block and remove from it a block that would contain the top piece of the pyramid. How big would that smaller block have to be? Well, if I’m cutting the pyramid exactly in half, the top piece would still have the same proportions as the original. Let’s say the original pyramid is 4 units wide by 12 units high. Then the top half would be 2 units wide by 6 units high. I can just remove a 2 by 6 block that would contain the top part. Ugh, that would be hard to cut out of the middle of the block. I’ll take it out of the top corner instead. There, I took a block that contained the original pyramid, and removed a block that contained the top half. 1/3 of the remaining shape is the bottom half of the pyramid. Now, what is the volume of the remaining clay? The bottom area is the square of the base, so 4 times 4. On the top surface we have 2 squared and 2 times 4. The height of each piece is 6, so the total is 6 times (16 + 4 + 8). Now take a third part of that, which is 2 times (16 + 4 + 8). That’s 56. I can see that it is right, because the volume of the original pyramid was 1/3 times 12 times 42, and I removed the top part with volume 1/3 times 6 times 22 . That is 64 – 12 = 56.
The method seems to work for all pyramids that are cut in half, but the resulting formula is general. That is, for a pyramid with base B and height kB (where k is some constant), we can take a block with base B^{2} and height kB, and remove from it a block with base b^{2} and height kb. The result of that is kB^{3} - kb^{3}, which is k(B-b)(B^{2}+ Bb +b^{2}). We know this from the difference of two cubes, but you can easily see that B^{3} - b^{3} = (B-b)(B^{2}+ Bb +b^{2}) by cutting a small cube out of a larger cube of clay. kB - kb is the height H of the cut-off pyramid. Then take 1/3 of it and you are done. Is it really that easy or did I make a mistake?