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Course: Geometry (FL B.E.S.T.)>Unit 9

Lesson 5: 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.

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• 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 there a simulator for this that I can see myself?
• 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.
• 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.
• Hello, what is that platform called that you are using.
• if the volume is the same does that mean that the surface area is also the same
• 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)
• 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?
• 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.
• 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.
• 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?
• So, from what I understand, you wouldn't have to cut the shape before skewing, as long as the material itself isn't stretching, as in, say, a ball of putty.

Here's another way to think about. Due to the way the principle works, there isn't a limit to how many "cuts" you can make, as Sal was essentially showing. So you could make an infinite amount of cuts, then skew it, but since the number of cross-sectional cuts is infinite, it would be the same as just skewing it without cutting, like you were describing.

The reason the volume doesn't change is because as another poster showed, the more you skew the shape, the diameter perpendicular to the slant edge of the shape grows smaller. Thus, while the surface area grows, the volume stays the same.

If you were to think about it in a 2D way, a parallelogram with a height of 5 and a base of 10 always has the same volume as a rectangle with the height of 5 and a base of 10, no matter how steep the angles are. I hope that helps!
(1 vote)
• Sal kept saying if they have the same area at 'any' point, then the have the same volume, but isn't it more correct to say if the same area at 'every' point? Maybe not that different but I was confused at first.