Measuring volume as area times length

I have this figure here. You could call it a rectangular prism. And I want to measure its volume. And I'm defining my unit cube as being a 1 centimeter by 1 centimeter by 1 centimeter cube. It has 1 centimeter width, 1 centimeter depth, 1 centimeter height. And I will call this, this is equal to 1 cubic centimeter. So I want to measure this volume in terms of cubic centimeters. We've already seen that we can do that by saying, hey, how many of these cubic centimeters can fit into this figure without them overlapping in any way? So if we had this in our hands, we could kind of try to go around it and try to count it, but it's hard to see here because there's some cubes that we can't see behind the ones that we are seeing. So I'm going to try different tactics at it. So first, let's just think about what we can observe. So we see that this one, if we measure its different dimensions, its width, it's 2 of the unit length wide. So it's 2 centimeters wide. It's 4 of our unit length-- we're defining our unit length as a centimeter-- it's 4 of our unit length high. So this dimension right over here is 4 centimeters. And it is 3 of our unit length deep. So this dimension right over here is 3 centimeters. So I want to explore if we can somehow use these numbers to figure out how many of these cubic centimeters would fit into this figure. And the first way I'm going to think about it is by looking at slices. So I'm going to take this slice right over here of our original figure. And let's think about how using these numbers, we can figure out how many unit cubes were in that slice. Well this is 2 centimeters wide and it is 4 centimeters high. And you might be saying, hey Sal, I could just count these things. I could get 8 squares here. But what if there was a ton there? It would be a lot harder. And you might realize well I could just multiply the width times the height, that would give me the area of this surface right over here. And it's only 1 deep so that also will give me the number of cubes. So let's do that. Let's find the area here. Well that's going to be 2 centimeters times 4 centimeters. That gives us the area of this. And then if we want to find out the number of cubes, well that's also going to be equivalent to the number of cubes. So we have 8 square centimeters is this area, and the number of cubes is 8. And if we want the number of cubes in the whole thing, we just have to multiply by the number of slices. And we see that we need one, two, three slices. This is 3 centimeters deep. So we're going to multiply that times 3. So we took the area of one surface. We took the area of this surface right over here. And then we multiply by the depth, that essentially gives us the number of cubes because the area of this surface gives us the number of cubes in an slice that is 1 cube deep. And then we would have to have 3 slices like that. So we would have to have this is 1 slice. We would have to have another slice, another slice and then another slice in order to construct the original figure. So 2 centimeters times 4 centimeters times 3 centimeters would give us our volume. Let's see if that works out. 2 times 4 is 8 times 3 is 24. Let me do that in that pink color. 24 centimeters cubed, or I could say cubic centimeters. So that's one way to measure the volume. Now there's multiple surfaces here. I happened to pick this surface, but I could have picked another one. I could have picked this surface right over here and done the exact same thing. So let's pick this surface and do the exact same thing. This surface is 3 centimeters by 4 centimeters. Let me do that in that blue color. Color changing is always difficult. So its area is going to be 12 square centimeters is the area of this surface. And 12 is also the number of cubes that we have in that slice. And so how many slices do we need like this in order to construct the original figure? Well we need, it's 2 centimeters deep. This is only 1 centimeter deep so we need two of them to construct the original figure. So we can essentially find the area of that first surface which was 3 times 4, and then multiply that times the width, times how many of those slices you need, so times 2. And once again, this is going to be 3 times 4 is 12 times 2 is 24. I didn't write the units this first time. But that's going to give us the count of how many cubic centimeters we have, how many unit cubes we can fit. So once again, this is 24 cubic centimeters. And you could imagine, you could do the same thing, not with this surface, not with this surface, but with the top surface. The top surface is 3 centimeters deep. And 2 centimeters wide. So you could view its area or its area is going to be 3 centimeters times 2 centimeters. So that area is-- let me do it in the same colors-- 3 centimeters times 2 centimeters which is 6 square centimeters. And that also tells you that there's going to be 6 cubes in this one cube deep slice. But how many of these slices do you need? Well you have this whole thing is 4 centimeters tall, and this thing is only 1 centimeter so you're going to need four of them. So that's 2, 3, try to draw it as neatly as I can, and 4. You're going to need 4 of these. So to figure out the whole volume, you going to have to take that and multiply that times 4 centimeters. So once again, 3 times 2 is 6 square centimeters times 4 centimeters is 24 cubic centimeters. So it doesn't matter what order you multiply these in. You could view this and take the area of one side then multiply it times the depth. Or you could take the surface area of another height and multiply it times the height or the width or the depth. And these are all the scenarios. But what it shows is that it doesn't matter what order we multiply these three dimensions in. You could take the 2 times the 4 first and then multiply it by the 3. Or you take the 3 times the 4 first and then multiply it by the 2. And or you could take the 2 times the 3 first, and then multiply by the 4. When you're multiplying, it doesn't matter what order you're doing these in. And so if you have a rectangular prism like this and you know it's three dimensions, you know it's 2 centimeters wide, 3 centimeters deep, and 4 centimeters tall, you could say, hey, the volume of this thing, the number of unit cubes, the number of cubic centimeters it can fit is going to be 2 centimeters times 4 centimeters times 3 centimeters, which we've seen three times already is 24 cubic centimeters.