Volumes of cones intuition | Solid geometry | High school geometry | Khan Academy

So I have two different three-dimensional figures here. I have a pyramid here on the left, and I have a cone here on the right. We know a few things about these two figures. First of all, they have the exact same height.

So this length right over here is h, and this length right over here, going from the peak to the center of the base here, is h as well. We also know that the area of the bases is the same. So, for example, in this left pyramid, the area of the base would be x times, and let's just assume that it is a square, so x times x. The area here is going to be equal to x squared.

The area of this base is equal to x squared, and the area of this base right over here would be equal to area is equal to pi times r squared. I'm saying that these two things are the same, so we also know that x squared is equal to pi r squared. Now, my question to you is: do these two figures have the same volume, or is it different? And if they are different, which one has a larger volume? Pause this video and try to think about that.

All right, now let's do this together. Now, given that we're talking about two figures that have the same height, and at least the area of the base is the same, you might be thinking that Cavalieri's principle might be useful. Just a reminder of what that is: Cavalieri's principle tells us that if you have two figures, and we're thinking in three dimensions, it tells us that if you 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 what we need to do is figure out if it is true that, at any point in this height, do these figures have the same cross-sectional area? Well, to think about that, let's pick an arbitrary point along this height. Just for simplicity, let's pick halfway along the height, although we could do this analysis at any point along the height.

So halfway along the height there, halfway along the height there; this distance right over here would be h over two, this distance right over here would be h over two. This whole thing is h. What we can do is construct what look like similar triangles, and we can even prove it to ourselves that these are similar triangles.

So let me construct them right over here. The reason why we know they're similar is that this line is going to be parallel to this line, and that this line is parallel to that line — to that radius. How do we know that? Well, we're taking cross-sectional areas that are parallel to the base, that are parallel to the surface on which it sits in this situation.

So in either case, these cross-sections are going to be parallel. These lines, which sit in these cross-sections or sit on the base and sit in the cross-section, have to be parallel as well. Well, because these are parallel lines, this angle is congruent to that angle. This angle is congruent to this angle because these are transversals across parallel lines, and these are just corresponding angles.

Of course, they share this angle in common, and here you see very clearly right angle right angle. This angle is congruent to that angle, and then both triangles share that. So the smaller triangle in either case is similar to the larger triangle. What that helps us realize is that the ratio between corresponding sides is going to be the same.

So if this side is h over 2 and the entire height is h, so this is half of the entire height, that tells us that this side is going to be half of r, so this right over here is going to be r over 2. This side over here, by the same argument, is going to be x over 2.

And so what's the cross-sectional area here? Well, it's going to be x over 2 squared, so it's going to be x over 2 squared, which is equal to x squared over 4, which is one-fourth of the base's area. What about over here? Well, this cross-sectional area is going to be pi times r over 2 squared, which is the same thing as pi r squared over 4.

Or we could say that is one-fourth pi r squared, which is the same thing as one-fourth of the area of the base. The area of the base is pi r squared. Now we're saying one-fourth pi r squared, so this is going to be equal to one-fourth the area.

We already said that these areas are the same, and so we've just seen that the cross-sectional area at that point of the height of both of these figures is the same. You could do that one-fourth along the height, three-fourths along the height; you're going to get the same exact analysis.

You're going to have two similar triangles, and you're going to see that you have the same areas, same cross-sectional areas at that point of the height. Therefore, we see by Cavalieri's principle, in three dimensions, that these two figures have the same volume.

What's interesting about that is it allows us to take the formula, which we've proven and gotten the intuition for in other videos, for the volume of a pyramid. We've learned that the volume of a pyramid is equal to one-third times base times height and say, "Well, this one must have the exact same volume; it must also be volumes equal to one-third times the area of the base times the height."

Because in both of these cases, the area of the base is the same and the height is the same, and we know that they have the same volume.