 Another quantity of interest is the volume of a three-dimensional object. And so, just as a recap, we've defined the length of a line sequin as the number of unit line sequins required to cover it. We've defined the area of a plane figure as the number of unit squares required to cover it. And so we're going to define the volume of a solid object in the same way. It's going to be the number of unit cubes that are required to cover it. Well, there's just one minor problem, which is that because we live in three dimensions, it's possible to cover a line segment or a plane figure with unit segments or unit squares. We can't actually do the same thing with solid objects. Covering it with a cube is not actually possible, but we can put it in a box and then define the volume of the solid object as the number of unit cubes that would fit into the box. So, for example, let's take a figure like this and we'll assume each one of these things is actually a unit cube. And I want to find the volume and, well, the volume is the number of unit cubes that we have. So now we have to go through this impossibly difficult mathematical task of counting cubes. Well, actually, it's pretty easy. The only problem we run into is that our limitation to two-dimensional representations of three-dimensional objects means that we'll have some cubes back here that we'll have to assume are complete cubes because we can't actually see them. All right, so we're going to count the number of cubes. And so let's see, that's one, two, three, four, five, six, and seven. So the volume is going to be seven cubic units. And that's all there is to volume. Everything else is implementation. Well, sometimes we can do some things that'll make this a little bit more efficient. So let's take a look at a rectangular prism, a box that's three units high, six units wide, and four units deep. Let's go ahead and find the volume. So let's actually draw a picture. Paper is cheap, so let's draw a picture. And we know that this prism is three units high, so I know that I can cut it into one, two, three slabs. Each of these is one unit thick. So here's our one, two, three units of height. The prism is six units wide, so I can cut it into six slabs, one unit thick. So here's one, two, three, four, five, six. So again, here's our width of six units. And then finally four units deep, which means I can cut this depth-wise into four units. So here's our one, two, three, four units of depth. So here's my three units high, six units wide, four units deep. And now I want to find the volume. Well, I need to count the number of cubes that are in this thing. Now, what I could do is I could count the cubes. One, two, three, four, five, six, and so on. And then I could do the same thing for each of the others. The problem is most of the cubes back here are going to be hidden by this front face. But because this is a prism, because it fills up the entire space, then we know that these last three slabs here are really copies of this front slab. So let's go ahead and take a look at that. Now, the prism is three cubes high. There's three cubes here. And it's six cubes wide. So that means I can figure out how many cubes there are in this, not by counting, one, two, three, four, not by counting, but it's three, six times. And so that means I have three times six cubes in that front slab. Now, I also know because the prism is four cubes deep, there's one, two, three, four of these slabs. So the total number of cubes, well, there's three by six in the front face, there's one, two, three, four of these slabs, so that's three by six by four cubes altogether. And at this point I can find the product, three times six times four is 72, and that is the volume of our rectangular prism.