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## Integrated math 2

### Course: Integrated math 2>Unit 10

Lesson 2: Cavalieri's principle and dissection methods

# Using related volumes

Use related volumes (using the volume of one figure to determine the volume of another). Created by Sal Khan.

## Video transcript

- [Instructor] We're told that all of the following figures have the same height. All of the figures except for B have square bases. So that's a square base, that's a square, that's a square, and that's a square. All of the figures except for C are prisms. Yeah, C is a pyramid right over here. All of the figures except for D are right. You can see D right over here is a little bit skewed, or you can view it as oblique. All of the figures except for E have the same base area. The base of figure E is a dilation of the base of figure A by a scale factor of 1.5. All right, so it tells a figure A has a volume of 28 cubic centimeters. What are the volumes of the other figures? So I'll pause this video and see if you can have a go with that. All right, now let's work through this together. Now they're telling us about the bases and the heights that a lot of these have the same base area, figure E's going to be different. And they also tell us they all have the same height. So one way to think about volume is it's going to deal with base and height. And so for figure A, it's pretty straight forward. If we call this area right over here, let's call that b, for the area of the base, and then it has some height, h, right over here. We know that the base area times the height is going to be the volume. So we can say that based on figure A, base times height is going to be equal to 28 cubic centimeters. Fair enough. Now what's going on over here with figure B? Well, it's a cylinder. Now for a cylinder, what is the volume of a cylinder? Well, it, too, is going to be base times height. So it's going to be the area of the base times the height. And if you're wondering, "How is that possible that it's the same as a volume of a rectangular prism over here?" It's actually Cavalieri's principle. If they have the same height and if at any point on that height, they have the same cross-sectional area, then you're going to have the same volume. So this volume is also going to be base times height. So let me just say this is figure A, figure B right over here, let me draw those dots a little better, these colons a little bit better. Figure B, the volume is also going to be base times height, which is equal to 28 cubic centimeters. Let me make that clear. That's the volume's equals to that. Volume is equal to that. Now what about for figure C? What is the volume going to be, what is the formula for the volume of a pyramid? And we've gotten the intuition and proven this to ourselves in other videos. Well, we know that for pyramid, the volume is going to be equal to 1/3 times base times height. And we know that it has the same base area as these other characters here, it has the same height. And so we know what base times height is. It's 28 cubic centimeters. So this is going to be 1/3 times 28 cubic centimeters. So this is going to be equal to 1/3 times 28 cubic centimeters, which we could rewrite as 28 over three cubic centimeters. You could also write that as nine and 1/3 cubic centimeters. So that's for figure C. Now let's think about figure D. I'll do that right over here. Well, for this oblique prism, I guess we could say, you're going to have the same idea that comes from Cavalieri's principle again. It's going to have the same formula for volume as figure A. It's going to be the base times the height right over here. So I could write volume is going to be equal to base times height and we already know what that is. They tell us. The base times the height is going to be the same as figure A. It's going to be 28 cubic centimeters. Now let's go to figure E. This is an interesting one 'cause it has a different base area. What is gonna be the area right over here? Well, they tell us that the base of figure E is a dilation of the base of figure A by a scale factor of 1.5. And these are both squares. So figure A, we'll say, x by x. This one over here is going to be 1.5x by 1.5x. So let me write that down. 1.5x times 1.5x. Or another way to think about it, let me do it over here where I have some free space. We know that b, which we know is an area of figure A, that would be equal to x times x. Now what's the area of the base of figure E? Well, it's going to be 1.5x times 1.5x or 1.5x squared, which is the same thing as 1.5 squared is 2.25x squared, and we know x squared or x times x that is equal to b. That is equal to our original base area in all of these other figures. So the area over here, this area right over here is going to be 2.25 times b. 2.25 times b. Now that wasn't so easy to read. Let me write that a little bit clearer. So 2.25b is the base area right over here. And so what's the volume of this figure? The volume is going to be the area of the base, which is 2.25 times the area of all these other figures' bases times the height, which is the same, times h. Now we know what base times, we know what b times h is. Where b is the area of figure, the base area of figure A. We know that b times h is 28 cubic centimeters. So the volume for figure E is going to be 2.25 times 28 cubic centimeters, times 28 cubic centimeters. And I don't have a calculator here in front of me and I can do it by hand, but I think you get the general point. You just have to multiply 2.25 times 28 to get the cubic, you get the volume of figure E. And that's because its base has been scaled in each dimension by 1.5.