Perimeter & area
What I want to do in this video is a fairly straightforward primer on perimeter and area. And I'll do perimeter here on the left, and I'll do area here on the right. And you're probably pretty familiar with these concepts, but we'll revisit it just in case you are not. Perimeter is essentially the distance to go around something or if you were to put a fence around something or if you were to measure-- if you were to put a tape around a figure, how long that tape would be. So for example, let's say I have a rectangle. And a rectangle is a figure that has 4 sides and 4 right angles. So this is a rectangle right here. I have 1, 2, 3, 4 right angles. And it has 4 sides, and the opposite sides are equal in length. So that side is going to be equal in length to that side, and that side is equal in length to that side. And maybe I'll label the points A, B, C, and D. And let's say we know the following. And we know that AB is equal to 7, and we know that BC is equal to 5. And we want to know, what is the perimeter of ABCD? So let me write it down. The perimeter of rectangle ABCD is just going to be equal to the sum of the lengths of the sides. If I were to build a fence, if this was like a plot of land, I would just have to measure-- how long is this side right over here? Well, we already know that's 7 in this color. So it's that side right over there is of length 7. So it'll be 7 plus this length over here, which is going to be 5. They tell us that. BC is 5. Plus 5. Plus DC is going to be the same length is AB, which is going to be 7 again. So plus 7. And then finally, DA a or AD, however you want to call it, is going to be the same length as BC, which is 5 again. So plus 5 again. So you have 7 plus 5 is 12 plus 7 plus 5 is 12 again. So you're going to have a perimeter of 24. And you could go the other way around. Let's say that you have a square, which is a special case of a rectangle. A square has 4 sides and 4 right angles, and all of the sides are equal. So let me draw a square here. My best attempt. So this is A, B, C, D. And we're going to tell ourselves that this right here is a square. And let's say that this square has a perimeter. So square has a perimeter of 36. So given that, what is the length of each of the sides? Well, all the sides are going to have the same length. Let's call them x. If AB is x, then BC is x, then DC is x, and AD is x. All of the sides are congruent. All of these segments are congruent. They all have the same measure, and we call that x. So if we want to figure out the perimeter here, it'll just be x plus x plus x plus x, or 4x. Let me write that. x plus x plus x plus x, which is equal to 4x, which is going to be equal to 36. They gave us that in the problem. And to solve this, 4 times something is 36, you could solve that probably in your head. But we could divide both sides by 4, and you get x is equal to 9. So this is a 9 by 9 square. This width is 9. This is 9, and then the height right over here is also 9. So that is perimeter. Area is kind of a measure of how much space does this thing take up in two dimensions? And one way to think about area is if I have a 1-by-1 square, so this is a 1-by-1 square-- and when I say 1-by-1, it means you only have to specify two dimensions for a square or a rectangle because the other two are going to be the same. So for example, you could call this a 5 by 7 rectangle because that immediately tells you, OK, this side is 5 and that side is 5. This side is 7, and that side is 7. And for a square, you could say it's a 1-by-1 square because that specifies all of the sides. You could really say, for a square, a square where on one side is 1, then really all the sides are going to be 1. So this is a 1-by-1 square. And so you can view the area of any figure as how many 1-by-1 squares can you fit on that figure? So for example, if we were going back to this rectangle right here, and I wanted to find out the area of this rectangle-- and the notation we can use for area is put something in brackets. So the area of rectangle ABCD is equal to the number of 1-by-1 squares we can fit on this rectangle. So let's try to do that just manually. I think you already might get a sense of how to do it a little bit quicker. But let's put a bunch of 1-by-1. So let's see. We have 5 1-by-1 squares this way and 7 this way. So I'm going to try my best to draw it neatly. So that's 1, 3, 3, 4, 5, 6, and then 7. 1, 2, 3, 4, 5, 6, 7. So going along one of the sides, if we just go along one of the sides like this, you could put 7 just along one side just like that. And then over here, how many can we see? We see that's 1 row. And that's 2 rows. Then we have 3 rows and then 4 rows and then 5 rows. 1, 2, 3, 4, 5. And that makes sense because this is 1, 1, 1, 1, 1. Should add up to 5. These are 1, 1, 1, 1, 1, 1, 1. Should add up to 7. Yup, there's 7. So this is 5 by 7. And then you could actually count these, and this is kind of straight forward multiplication. If you want to know the total number of cubes here, you could count it, or you can say, well, I've got 5 rows, 7 columns. I'm going to have 35-- did I say cube-- squares. I have 5 squares in this direction, 7 in this direction. So I'm going to have 35 total squares. So the area of this figure right over here is 35. And so the general method, you could just say, well, I'm just going to take one of the dimensions and multiply it by the other dimension. So if I have a rectangle, let's say the rectangle is 1/2 by 1/2 by 2. Those are its dimensions. Well, you could just multiply it. You say 1/2 times 2. The area here is going to be 1. And you might say, well, what does 1/2 mean? Well, it means, in this dimension, I could only fit 1/2 of a 1-by-1 square. So if I wanted to do the whole 1-by-1 square, it's all distorted here. It would look like that. So I'm only doing half of one. I'm doing another half of one just like that. And so when you add this guy and this guy together, you are going to get a whole one. Now what about area of a square? Well, a square is just a special case where the length and the width are the same. So if I have a square-- let me draw a square here. And let's call that XYZ-- I don't know, let's make this S. And let's say I wanted to find the area and let's say I know one side over here is 2. So XS is equal to 2, and I want to find the area of XYZS. So once again, I use the brackets to specify the area of this figure, of this polygon right here, this square. And we know it's a square. We know all the sides are equal. Well, it's a special case of a rectangle where we would multiply the length times the width. We know that they're the same thing. If this is 2, then this is going to be 2. So you just multiply 2 times 2. Or if you want to think of it, you square it, which is where the word comes from-- squaring something. So you multiply 2 times 2, which is equal to 2 squared. That's where the word comes from, finding the area of a square, which is equal to 4. And you could see that you could easily fit 4 1-by-1 squares on this 2-by-2 square.