By Vanderah T.A.(ed)

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SANDRA. One third abh. SERGE LANG. Yes, so we write Volume of pyramid But that is the most general theorem! With this method, I take the pyramids with the top at the center of the cube, and I get six pyramids. In this situation, the simplest case was of a pyramid with a base of sides I but height 1/2. Then I could get the volume directly, for a pyramid like the Egyptian pyramids, when the top lies directly above the center of the base. SERGE. I think you could get it a different way. There are three pyramids which make up the cube.

R to the fourth, times V. SERGE LANG. r4V, where V is the 4-dimensional volume. And suppose I have a solid in n dimensions, and I make a dilation by a factor of r in all directions, in all n dimensions. How does the volume change? SERGE. r to the power n. SERGE LANG. rn, and that's how it is in n dimensions. OK? Any prob- lems? Sandra. SANDRA. No. ] SERGE LANG. All right. So we see how volume changes in n dimenSlOns. Let's go back to three dimensions, and see how we can find the volume of some standard figures.

I start with the pyramid with height 112. The height of the new pyramid is I. By how much have I dilated in this direction? MIKE. It's doubled. SERGE LANG. Certainly, so I have dilated by what factor? MIKE. 2. SERGE LANG. That's right. So I have three pyramids. The first with sides I, I and height 112. The second with sides a, b and height 112. The third with sides a, b and height I, which is obtained from the second by dilation by a factor of 2 in one direction. The volume of the second pyramid is ~ abo What is the volume of the third pyramid~' MIKE.

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