Perimeter and Area

[Single Speaker]

Narrator: What I wanna do in this video is a pretty straight forward primer on perimeter and area. I’ll do perimeter on the left and I’ll do area here on the right [Writes Perimeter on the left side of the screen and area 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 measure, or 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. [Draws a rectangle and puts right angles in each of the corners of the rectangle.] It has one, two, three, four right angles and it has four sides and the opposite sides are equal in length. So that side in length is gonna be equal to that side. [Shows how the bottom side of the rectangle is equal in length to the top side.] And that side is equal in length to that side. [Shows how the left side of the rectangle is equal in length to the right side.] And maybe I’ll label the points, A, B, C, and D. [Labels the corners of the rectangle in clockwise order A-d starting with the top left corner.] And let’s say we know the following: we know that AB is equal to 7 and we know that BC is equal to 5. [Beneath the rectangle he writes AB=7 and BC=5.] And we want to know what is the perimeter of ABCD? [Beneath the lengths he wrote he writes “Perimeter of ABCD=”]

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? [Side AB.] Well we already know that’s 7. So that side right over there is of length 7. [Labels the top side of the rectangle 7.] So it’ll be 7 plus this length over here which is going to be 5. [Labels the side BC 5.] Plus DC is going to be the same length as AB which is going to be 7 again. [Labels DC 7.] Then finally DA or AD however you want to call it, is going to be the same length         as BC, which is 5 again. So you have 7, plus 5, is 12, plus 7, plus 5 is 12 again. [At the bottom of the screen he has written 7+5+7+5 after Perimeter of ABCD.] So you’re going to have a perimeter of 24. [Writes =24 at the bottom of the screen.] 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. [Draws a square beneath the rectangle example he made.] My best attempt. So this is A, B, C, D. [Labels the corners of the square the same as the rectangle, beginning in the tope left corner, A-D.] And we’re going to tell ourselves that this right here is a square. [Labels the square, square.] And let’s say that this square has a perimeter of 36. So given that, what is the length of each of these sides? Well all the sides are going to have the same length, let’s call them x. So if AB is X, then BC is X, then DC is X, and AD is X. All the sides are congruent, all of these segments are congruent, they all have the same measure, we call that X. [Labels side AB x then marks the other sides with a hash mark to show they are the same.] So if we wanna figure out the perimeter here, it’ll just be x plus x plus x plus x, or 4x. Let me write that. [Writes x+x+x+x=4x on the screen.] Which is going to be equal to 36. [Writes =36 at the end of the equation.] They gave us that at the end of 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. [Writes X=9 on the screen.] So this is a 9 by 9 square. This width is 9. [Referring to the sides of the square.] 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? [Referring to the first rectangle.] And one way to think about area is if I have a 1 by 1 square [draws a square with each side being 1 unit long] 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, [referring to the first rectangle he drew under perimeter] you could call this a 5 by 7 rectangle because that immediately tells you, okay, this side is 5 and that side is 5 [referring to the two short parallel sides of the rectangle]. This side is 7 and that side is 7 [referring to the parallel long sides of the rectangle]. 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. [Writes 1x1 square on the screen next to the 1 unit square he drew on the screen.] 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, [the first rectangle he drew under perimeter] and I wanted to find out the area of this rectangle -- and the notation we can use for area is put something in brackets. [Writes brackets on the screen.] So the area of rectangle ABCD [writes ABCD in the brackets] 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 might already 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. [Begins drawing 35 squares in the rectangle.] We have 5 1-by-1 squares this way and 7 this way, so I’m gonna try my best to draw it neatly. [He draws the squares by dividing the rectangle lengthwise into 7 sections with 6 lines, then width wise into 5 sections with 4 lines. Labels the length of the 35 unit squares as 1, indicating that the rectangle is 7 unit squares in length and 5 unit squares in width.] So this is 5 by 7, 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 gonna have 35 total squares.

So the area of this figure right over here is 35. [“[ABCD]=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 ½ by 2. Those are its dimensions. Well you could just multiply. You say ½ times 2. The area here is going to be 1. And you might say, well, what does ½ mean? Well, it means, in this dimension, I could only fit ½ of a 1-by-1 square. So if I wanted to do the whole 1-by-1 square, it’s a little distorted here. [Draws a unit square over the rectangle he drew, but the width of the rectangle is only half the width of a 1 unit square.] It would look like that. So I’m only doing half of one. I’m doing another half of one just like that. [Draws another unit square over the other half of the rectangle, next to the first unit square he drew over the rectangle.] 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. [Draws a square on the screen.] And let’s call that XYZ-- I don’t know, let’s make this S. [Labels the points on the screen ] 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. [Writes [XYZS] on the screen.] 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 [referring to one of the sides], then this is going to be 2 [referring to another side.] 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. 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. [Divides the square into four little squares.]

[End of Video]