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## High school geometry

### Course: High school geometry>Unit 9

Lesson 2: Cavalieri's principle and dissection methods

# Cavalieri's principle in 3D

Cavalieri's principle tells us that if 2 figures have the same height and the same cross-sectional area at every point along that height, they have the same volume. Created by Sal Khan.

## Want to join the conversation?

• If the volume is the same, does that mean that the surface area is also the same?
• No. Just like how you can have a 9x4 rectangle and a 12x3 rectangle with the same area but different perimeter, it's completely possible to have solids with the same volume and different surface area, and vice versa.
• Is it true that in order for 2 spheres to satisfy Cavalieri's principle, the only constraint is for them to have the same radius?
• If two spheres have the same radius, they are both going to be the same size and shape, and therefore they would both have the same volume.
• Is there a simulator for this that I can see myself?
(1 vote)
• Can the figure be infinitely skewed? I'm having trouble imagining the following:

Two identical cylinders, A and B, with same height and same base. Then we skew the bottom of cylinder A to the right by a mile. Compared to cylinder B, cylinder A has the same height as B and both have the same cross sectional area along that height, so they still meet Cavalieri's Principle and thus have the same volume.

Yet, my intuition says that cylinder A, which has been skewed by a mile, could hold way more water inside itself as cylinder B, so it should have a greater volume!

How to solve that apparent contradiction?
• Yes, it may seem a little counter-intuitive, but the more you skew the cylinder, the thinner it gets, and the skew has no bearing on the actual volume.
• if the volume is the same does that mean that the surface area is also the same
(1 vote)
• Well if the volume is the same, it doesn’t necessarily mean the surface area has to be the same. To keep it simple, use this example:

1. You have 2 rectangles (one named “A” and one named “B”

2. The Area Of Rectangle A and B is 100

3. The 2 side lengths of Rectangle A is 4 and 25

4. The 2 side lengths of Rectangle B is 10 and 10

5. Both have the same area, but different Perimeters. Rectangle A has a perimeter of 58, and Rectangle B has a perimeter of 40.

6. Therefore they have same area but different perimeters.

7. This is the same relationship as between Volume (as area) and Surface Area (as Perimeter)
• 2, 3, 7, 23, then a humongous number that I don't care to count. 2, 6, a humongous number that I don't care to count. 2, 9, 2, a humongous number that I don't care to count. 2, 5, 21+, a humongous number that I don't care to count.
(1 vote)
• So technically if you were to skew two of the same exact shapes, you would have to cut it into chips. You couldn't just stretch it to the right or left without slicing it? I was wondering, because if you skewed it to the right without cutting it, wouldn't that increase the volume?

Say you have two cylinders of the same height and cross-sectional area. You fill them with water. If you stretch the one on the right to the right, without cutting it, wouldn't it increase the volume?

Essentially, my question is, in order to use Cavalieri's Principle, the shapes must be cut before skewed?
(1 vote)
• Instead of the same cross-section wouldn't it be simpler to just say the same base?
(1 vote)
• I do not think so, base has a very strict definition for prisms, so cross-sections are somewhere between the two bases of the prism. So the area of any horizontal cross section would equal the area of the base, but it is not a base.
(1 vote)
• If two figures have the same height, the same cross-sectional area at every point along that height, and the same volume, then they are the exact same figure, right?
(1 vote)
• Not necessarily. Cavalieri's principle is that the shapes can vary, but the properties will remain intact. For example, if you stack 4 quarters on top of each other, they would be the same height, volume, and cross sectional area, but not be the same exact shape.
(1 vote)
• Hello, what is that platform called that you are using.
(1 vote)

## Video transcript

- [Instructor] So we have two cylinders here. Let's say we know that they have the exact same volume and that makes sense because it looks like they have the same area of their base and they have the same height. Now what I'm going to do is start cutting up this left cylinder here and shifting things around. So if I just cut it in two and take that bottom cylinder, that bottom half and shift it a bit, have I changed its volume? Well, clearly I have not changed its volume. I still have the same volume. The combined volume of both of these half cylinders, I could say, are equal to the original cylinder. Now what if I were to cut it up even more? So let me cut it up now into three. Well, once again I still haven't changed my original volume. It's still the same volume as original and I just cut it up into thirds. And if I shift them around a little bit I'm not changing the volume. And I could keep doing that. I could cut it up into a bunch of them. Notice, this still has the same original volume, I've just cut it up into a bunch of sections. I've cut it horizontally and now I'm just shifting things around, but that doesn't change the volume. And I can do it a bunch of times. This looks like some type of poker chips or gambling chips where I can have my original cylinder and now I've cut it horizontally into a bunch of these, I guess you could say chips but clearly it has the same combined volume. I can shift it around a bit but it has the same volume. And this leads us to an interesting question and it's actually a principle known as Cavalieri's principle, which is if I have two figures that have the same height and at any point along that height, the cross-sectional area is the same, then the two figures have the same volume. Now how does what I just say apply to what's going on here? Well, clearly both of these figures have the same height and then at any point here, wherever I did the cuts, at the same point on this original cylinder, well, my cross-sectional area is going to be the same because it's going to be the same area as the base in the case of this cylinder and so it meets Cavalieri's principle. But Cavalieri's principle's nothing exotic. It comes straight out of common sense. I can just do more cuts like this and you can see that I have, you can see a more continuous looking skewed cylinder but this will have the same volume as our original cylinder. When I shift it around like this, it's not changing the volume. And that's not just true for cylinders. I could do the exact same argument with some form of a prism. Once again they have the same volume. I could shift, I could cut the left one in half and shift it around, doesn't change its volume. I could cut it more and shift those around, still doesn't change the volume. So Cavalieri's principle seems to make a lot of intuitive sense here. If I have two figures that have the same height and at any point along that height, the cross-sectional area is the same, then the figures have the same volume. So these figures also have the same volume. And I could do it with interesting things like, say, a pyramid. These two pyramids have the same volume and I were to cut the left pyramid halfway along its height and shift the bottom like this, that doesn't change its volume. And I can keep doing that with more and more cuts. And 'cause at any point here, these figures have the same height and at any point on that height, the cross-sectional area is the same, and so they have the same volume. But once again it is intuitive. And it goes all the way to the case where you have, you could view it as a continuous pyramid right over here that has been skewed. So no matter how much you skew it, it's gonna have the same volume as our original pyramid 'cause they have the same height. And the cross-sectional area at any point in the height is going to be the same. We can actually do this with any figure. So these spheres have the same volume. I could cut the left one in half, halfway along its height and shift it like this. Clearly, I'm not changing the volume. And I could make more cuts like that. And clearly it has still the same volume. And this meets Cavalieri's principle because they have the same height and the cross section at any point along that height is going to be the same. So even though I can cut that one up and I can shift it, it looks like a different type of object, a different type of thing, but they have the same height and cross sections at any point are the same area, so we have the same volume, which is a useful thing to know not just to know the principle but hopefully this video helps you gain some of the intuition for why it makes intuitive sense.