Volume of pyramids intuition | Solid geometry | High school geometry | Khan Academy

In this video, we're going to talk about the volume of a pyramid. Many of you might already be familiar with the formula for the volume of a pyramid, but the goal of this video is to give us an intuition or to get us some arguments as to why that is the formula for the volume of a pyramid.

So let's just start by drawing ourselves a pyramid. I'll draw one with a rectangular base, but depending on how we look at the formula, we could have a more general version. A pyramid looks something like this, and you might get a sense of what the formula for the volume of a pyramid might be if we say this dimension right over here is X, this dimension right over here—the length—is Y, and then you have a height of this pyramid. If you were to go from the center straight to the top or you were to measure this distance right over here, which is the height of the pyramid, let’s call that Z.

You might say, “Well, I’m dealing with three dimensions, so maybe I multiply the three dimensions together, and that would give you volume in terms of units.” But if you just multiplied XY * Z, that would give you the volume of the entire rectangular prism that contains the pyramid. So that would give you the volume of this thing, which is clearly bigger and has a larger volume than the pyramid itself. The pyramid is fully contained inside of it.

This would be the tip of the pyramid on the surface, just like that. You might get a sense that, all right, maybe the volume of the pyramid is equal to X * Y * Z times some constant. What we're going to do in this video is have an argument as to what that constant should be, assuming that this is the volume of the pyramid and to help us with that.

Let's draw a larger rectangular prism and break it up into six pyramids that completely make up the volume of the rectangular prism. So first, let’s imagine a pyramid that looks something like this, where its width is X, its depth is Y, so that could be its base, and its height is halfway up the rectangular prism. If the rectangular prism has a height of Z, the pyramid's height is going to be Z over 2.

Now, what would be the volume of that pyramid based on what we just saw over here? Well, that volume would be equal to some constant K times X * Y, not times Z, times the height of the pyramid times Z/2. So it would be X * Y * Z/2. I'll just write times Z over two. Or actually, we could even write it this way: XY * Z over 2.

Now I can construct another pyramid that has the exact same dimensions. If I were to just flip that existing pyramid on its head, it would look something like this. This pyramid also has dimensions of X width, Y depth, and a Z over two height. So its volume would be this as well. Now, what is the combined volume of these two pyramids? Well, it’s just going to be this times two.

So the combined volume of these pyramids—let me just draw it that way—so these two pyramids that look something like this, I’m going to try to color code it. We have two of them, so two times their volume is going to be equal to K * X * Y * Z, K * X * Y, and Z.

We have more pyramids to deal with. For example, I have this pyramid right over here where this face is its base, and then if I try to draw the pyramid, it looks something like this. This one right over there—now what is its volume going to be? Its volume is going to be equal to K times its base, which is Y * Z. So K * Y * Z. What’s its height? Well, its height is going to be half of X, so this height right over here is half of X.

So it’s K * Y * Z * X over 2, or I could say times X and then divide everything by two. Now I have another pyramid that has the exact same dimensions. This one over here, if I try to draw it on the other face opposite the one we just saw—essentially, if we just flip this one over—it has the exact same dimensions.

So one way to think about it is that we have two pyramids that look like that with those types of dimensions. This is for an arbitrary rectangular prism that we are dealing with. So I have two of these, and if you have two of their volumes, what’s it going to be? It’s just going to be two times this expression, so it’s going to be K times X * Y * Z, X * Y * Z—interesting!

Then last but not least, we have two more pyramids. We have this one that has a face that has the base right over here—that’s its base—and if it was transparent, you’d be able to see where I’m drawing right here. Then you have one on the opposite side right over there on the other side, and like I said, if you were to flip this around.

By the exact same argument—let me just draw it—so we have two of these pyramids. I’ll do my best to draw it. So times two. Each of them would have a volume of what? Each of them, their base is X * Z, so it’s going to be K * X * Z—that’s the area of their base. And then what is their height? Well, each of them has a height of Y over two, so times Y over 2.

I have two of those pyramids, so I’m going to multiply those by two. The twos cancel out, so I’m just left with K * X * Y * Z. So K times X * Y * Z. Now one of the interesting things that we’ve just stumbled on in this is seeing that, even though these pyramids have different dimensions and look different, they all have actually the same volume, which is interesting in and of itself.

So if we were to add up the volumes of all of the pyramids here and use this formula to express them, if I were to add all of them together, that should be equal to the volume of the entire rectangular prism. Maybe we can figure out K. So the volume of the entire rectangular prism is X * Y * Z.

X * Y * Z, and then that’s got to be equal to the sum of these. So that’s going to be equal to K * X * Y * Z plus K * X * Y * Z plus K * X * Y * Z. Or you could say that’s going to be equal to 3K * X * Y * Z. All I did was say let me just add up the volume from all of these pyramids.

So what do we get for K? Well, we could divide both sides by 3 * X * Y * Z to solve for K. 3 * X * Y * Z divided by 3 * X * Y * Z, and we are left with, on the right-hand side, everything cancels out. We're just left with K, and on the left-hand side, we're left with 1/3.

So we get K is equal to 1/3. K is equal to 1/3, and there you have it—that's our argument for why the volume of a pyramid is 1/3 times the dimensions of the base times the height. You might see it written that way, or you might see it written as 1/3 times base, and so if X * Y is the base, so the area of the base, so the base area times the height, which in this case is Z.

But if you say H for that, you might see the formula for a pyramid written this way as well, but they are equivalent. That’s why you should feel good about the 1/3 part.