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Volume of pyramids intuition

The volume of a pyramid is a fraction of the volume of the rectangular prism that encloses it. We can find out what that fraction is by cutting a prism into several pyramids. Created by Sal Khan.

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Video transcript

- [Instructor] In this video, we're going to talk about the volume of a pyramid. And 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. And I'll draw one with a rectangular base. But depending on how we look at the formula, we could have a more general version. But 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 right over here is y. And then you have a height of this pyramid. If you were to go from the center straight to the top or if you were to measure this distance right over here, which is the height of the pyramid. You'll just call that, let's call that z. And so you might say well, I'm dealing with three dimensions, so maybe I'll multiply the three dimensions together and that would give you volume in terms of units. But if you just multiplied xy times z, that would give volume of the entire rectangular prism that contains the pyramid. So that would give you the volume of this thing, which is clearly bigger, has a larger volume than the pyramid itself. The pyramid is fully contained inside of it. So this would be the tip of the pyramid on the surface, it's just like that. And so you might get a sense that, all right maybe the volume of the pyramid is equal to x times y times z, times some constant. And what we're going to do in this video is have an argument as to what that constant should be. Assuming that this, the volume of the parameter is roughly of the structure. 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. So the rectangular prism has height z, the pyramid's height is going to be z over two. Now what would be the volume of the pyramid based on what we just saw over here? Well, that value would be equal to some constant k times x, times y, not times z, times the height of the pyramid, times z over two. So it'd be x times y times z over two, I'll just write times z over two or actually we can even write it this way xy is z over two. Now I can construct another pyramid has the exact same dimensions. If I were to just flip that existing pyramid on its head and look something like this. This pyramid also has dimensions of an x width, a y depth and a z over two height. So it's 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 gonna try to color code it. We have two of them. So two times their volume, is going to be equal to well two times this is just going to be k times xyz. Kxy and z. And 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 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 is y times z so kyz. And 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 times y times z times x over two 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 essential if we just flip this one over, has the exact same dimensions. So one way to think about it, 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 so 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 xyz. xyz, interesting. And 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. And then you have one on the opposite side, right over, there on the other side. Like as if you were to flip this around. And so by the exact same argument, so let me just draw it. So we have two of these, two of these pyramids my best to draw it so times two. So each of them would have a volume of what? Each of them their base is x times z. So it's going to be k times x times 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 two and 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 times xyz. So k times xyz. 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 themselves. And so if we were to add up the volumes of all of the pyramids here and use this formula to express them, so if I were to add all of them together that should be equal to the volume of the entire rectangular prism. And then maybe we can figure out k. So the volume of the entire rectangular prism is xyz. X times y times z and then that's got to be equal to the sum of these. So that's going to be equal to kxyz plus kxyz plus kxyz or you could say that's going to be equal to three kxyz. All I did is, let me just add up the volume from all of these pyramids. And so what do we get for k? Well, we could divide both sides by three xyz to solve for k, three xyz. Three xyz and we are left with on the right hand side the everything cancels out we're just left with a k. And on the left hand side we're left with a 1/3. And 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. So you might see it written that way or you might see it written as 1/3 times base and so if x times 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, but that's why you should feel good about the 1/3 part.