Cavalieri's principle in 3D | Solid geometry | High school geometry | Khan Academy

So we have two cylinders here, and 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 the original. Now, just cut it up into thirds, and if I shift them around a little bit, I'm not changing the volume.

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—a bunch. I've cut it horizontally, and now I'm just shifting things around, but that doesn't change the volume. I can do it a bunch of times. This looks like some type of poker chips or gambling chips, where I could 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, both clearly—both of these figures have the same height, and then at any point here, wherever I did the cuts at this point, 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 is 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 could say, 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 cut the left one in half and shift it around; it doesn't change its volume.

I could cut it more and shift those around; it 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 if 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 because 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 volumes. 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 going to have the same volume as our original pyramid because 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 the 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.