Special polyhedra: cube

0:00- [Instructor] What we're gonna do in this video is think about how to find the volume of a cube. But before we do that, I wanna think about or maybe review how we find the area of a square. So let me draw a square here, and I'm kind of hand drawing it so it's not going to be perfect, but that one's looking very square-like.

0:22And we know that what makes it a square is that the sides are all the same length. And we also know that these are all right angles and they're all the same length. And that's what makes it a square. Now, let's say that this square, one of the sides has length of three units.

0:42Well, that means that all of the sides have length of three units. And this is a bit of review, but what would be the area of this square? Well, we could think about it this way. It's three units, it this direction, 1, 2, 3. And that looks about right and let me make it roughly equal.

1:01So that looks about right. And it's also three units in this direction. Let me make it look a little bit closer. And so what we could do is we could create a grid. So if we are three units in this direction, so let me split this into three and then three in this direction, we split here, three, we can see that we have, there's a bunch of ways you could think about it.

1:29You could have 1, 2, 3 rows, and each of those rows have three units in them, three square units. So the area here is going to be three rows of three is equal to nine. And our units would be square units, nine units squared. Or another way to think about it, you could say area is equal to three to the second power units squared.

1:59Or another way to think about it is when you see something with an exponent of two people often say three squared. Why do they say that? Well, that's 'cause that's how you find the area of a square. So let's keep that in the back of our mind to think about the volume of a cube.

2:15And so let me actually, let me make a cube. That is that each of its sides are three units. And to do that, I am just going to copy and paste this character that we have right here. Let me just copy, let me just do this, and I'll build from there. So I'm gonna copy and then I'm gonna paste it.

2:40Let me put it right over there. And now let me make a cube out of it. And so what do I mean by that? Well, we know it's three wide, three high, but let me also make it three deep. So three deep, just like I'm trying to make it look like a real cube. I can do a better job than that.

3:00Okay, that looks pretty good. Three deep. And then let me make this right over here. And then, okay, that looks pretty close to a cube. So what is going to be the volume of this cube? And one way to think about it, area was how many of these unit squares could we fit inside of it?

3:20And we saw we had 1, 2, 3, 4, 5, 6, 7, 8, 9. When we're thinking about volume, we're thinking about how many of these unit cubes can we fit in here? And this is one of the unit cubes right over here that I am showing a little bit. That's one of them. And so you could try to visualize and think about how to count it, or we can do a very similar idea to what we did before.

3:45Because this is a cube. All the dimensions are three. So there's three wide, three high, we already saw that, but it's also three deep. So let me make this so three deep. So let me divide it like that. And then we have this over here. I haven't drawn it as nicely as one could, but you get the idea.

4:11Give me some grace that this is a hand-drawn, this is a hand-drawn example right over here. Maybe has a little bit of depth, a little bit of perspective. You can see that. That didn't help, that just made it a little bit messier. But what you can see here is we essentially have, on this, let me divide it, so we have three layers deep.

4:33We have this first layer of cubes that I'm going to highlight in red. So we have these in red. How many of those are there? So in this layer, well you could see, we can count 1, 2, 3, 4, 5, 6, 7, 8, 9. Or you could say there's three times three there. And it's, it doesn't go deeper than one layer.

4:56So it on this one right over here, we only have nine. But then how many of these layers do we have? We have that one. Then we have another one right over here in orange. That's the second layer. And then the third layer we have, I'll make that in blue right over here.

5:16I'll color that third layer in blue so you can see it so it doesn't get too quenched up. And I'll color that middle layer in orange so that you can see it. So you can see there's three layers. So I'm going to multiply that by another three. And so how many do you get?

5:35You get three times three is equal to 27 of these unit cubes or cubic units unit. Unit cubed. And actually you could, well, you could write it like that. Or actually, let me just get rid of, I think you should actually say square units or cubic units, just like that.

5:57If this was a meter, you would say meters cubed or a cubic meter. And what you can see here is, what's the same thing as three times three times three? That's the same thing as three to the third power. And you can see that that's why people call this three cubed because that's how you find the volume of a cube.

6:16So if you give me the length of any side of a cube, we know that all of them are the same. So we'll know that the volume is going to be equal to if it's the length, times the length, times the length, which is equal to the length cubed. If someone gives you the height, well, that's going be the same thing as the length.

6:33It's gonna be the height times the height times the height cubed. Whatever side they give you... Or sorry, times the height, which is equal to height cubed. So whatever side they give you, you just multiply it three times or you can take three of them and multiply 'em together is one way to say it.

6:50Or you can just cube it.