I've got a square here. It's right like that. What makes it a square is all of the sides are equal. I haven't gone in depth into angles yet, but these are at right angles to each other. I'll just draw it like that. That means that if this bottom side goes straight left and right, that this left side will go straight up and down. That's all the right angle really means. Let's say that the side down here is equal to 8 meters. This side right here. And this is a square. And I were to ask you what is the area of the square? Well, the area is essentially how much space the square takes up, let's say, on your screen right now. So it's essentially a way of measuring how much space something takes up on kind of a two-dimensional surface. A two-dimensional surface would just be this computer screen or your piece of paper, if you're also doing this problem. An analogy would be if you had an 8 meter by 8 meter room, how much carpeting would you need is kind of the size of the space you need to fill out in two dimensions on some type of surface. So the area here is literally how much is this size that you're filling up, and it's very easy to figure out for a square. It's literally going to be your base times your height -- and this is true for any rectangle -- but since this is a square, your base and your height are going to be the same number. It's going to be 8 meters. So your area is going to be 8 meters times 8 meters, which is equal to 8 times 8 is 64, and then your meters times your meters -- you have to do the same thing with your units -- you get 64 meters squared. Or another way of saying, this is 64 square meters. You might be asking where are those 64 square meters? Well, you can actually break it out here. So let me draw it a little bit bigger than I originally drew it. I probably should have drawn it this big to begin with. So let's say that's my same square. I'm going to draw a little bit, so let me divide it in the middle. Let me see, I have -- and we divide them again. Then we divide each side again just like that. I could probably do it neater. And let me do it one more time. Divide these just like that, and then divide these just like that. There you go. OK. Now the reason why I did this is to show you the dimensions along the base and the height. We said this is 8 meters, and notice I have 1, 2, 3, 4, 5, 6, 7, 8 meters. And the same thing along this side. 1, 2, 3, 4, 5, 6, 7, 8 meters. So when we're talking about 64 square meters, we're literally counting each of the square meters. A square meter is a two-dimensional measurement, that's 1 meter on each side. That's 1 meter, that's 1 meter. What I'm shading here in yellow is 1 square meter. And you could imagine just counting the square meters. In each row we're going to have 1, 2, 3, 4, 5, 6, 7, 8 square meters. And then we have 8 rows. So we're going to have 8 times 8 square meters or 64 meters square. Which is essentially if you sat here and just counted each of these, you would count 64 square meters. Now, what happens if I were to ask you the perimeter of my square? The perimeter is the distance you need to go to go around the square. It's not measuring, for example, how much carpeting you need. It's measuring, for example, if you wanted to put a fence around your carpet -- I'm kind of mixing the indoor and outdoor analogies -- it would be how much fencing you would need. So it would be the distance around. So it would be that distance plus that distance plus that distance plus that distance. But we already know this distance right here on the bottom, we already know this distance is 8 meters. Then we know that the height right here is 8 meters. It's a square. This distance up here is going to be the same as this distance down here -- it's going to be another 8 meters. Then when you go down the left hand side it's going to be another 8 meters. We have four sides -- 1, 2, 3, 4 -- each of them are 8 meters. So you add 8 to itself 4 times, that's the same thing as 8 times 4, you get 36 meters. Now notice, when we measured just the amount of fencing we needed, we ended up just with meters, just with kind of a one-dimensional measurement. That's because we're not measuring square meters here. We're not measuring how much area we're taking up. We're measuring a distance -- a distance to go around. We are taking turns, but you can imagine straightening out this fence, and it would just become one big fence like this, which would have the same length of 36 meters. So that's why we just have meters there for perimeter. But for area we got square meters, because we're counting these two-dimensional measurements. Now, let's make it a little bit more interesting. What happens if instead of a square I have a rectangle like this? Let's say that this side over here is 7 centimeters. And let's say that the height right here is 4 centimeters. So what is the area of this rectangle going to be? It's going to be 7 times 4 centimeters. 7 centimeters times 4 centimeters. Remember, we could draw 7 rows, right, and each of them is going to have 4 square centimeters -- each of those is a square centimeter. So if you were to count them all out, you'd have 7 times 4 square centimeters. It's 4 centimeters. So it's equal to 28 centimeters square or squared centimeters. What's the perimeter? Well, it's going to be equal to this distance down here, which is 7 centimeters, plus this distance over here which is 4 centimeters, plus the distance on the top -- this is a rectangle, it's going to be the same distance as this one over here. So plus another 7 centimeters. Then you're going to have this distance on the left hand side. But this distance on the left hand side is the same as this distance right here -- this is also 4 centimeters. So plus another 4 centimeters. And what do you get? You get 7 plus 4 which is 11, and then you have another 7 plus 4. You have 11 plus 11, so you have 22 centimeters. Once again, it's not a square centimeter. Now let's divert -- let's go away from our rectangle analogy or our rectangle examples. So let's see if we can do the same with triangles. So let's say I have a triangle here. I have a triangle like this. Let's say that this distance right here -- actually let me draw it like this. I think this'll make it a little bit easier for you to see how this relates to a rectangle. Let me draw it like this. There you go. That's my triangle. And let's say that this distance right here is 7 centimeters right down there. And let's say that the height of this triangle is 4 centimeters. And I were to ask you what is the area of the triangle? Well, when we had a rectangle like this, we just multiplied 7 times 4. But what would that give us? That would give us the area of an entire rectangle. If we did 7 times 4, that would give us the area of this entire rectangle. You could imagine extending my triangle up like this. This is a right triangle -- this is going straight up and down, this is going straight left and right on the bottom right here. It's a 90 degree angle, if you've been exposed to the idea of angles already. So you could almost view it as it's 1/2 of this rectangle. Not really almost, it is. Because if you just double this guy, you could imagine if you flip this triangle over, you get the same triangle but it's just upside down and flipped over. So if you think about when you multiply 7 times 4, you're getting the area of this entire rectangle, which we just did up here. But we want to know the area of the triangle. We want to know just this area right here. You can see, hopefully, from this drawing that the area of this triangle is exactly 1/2 of the area of the entire rectangle. So the area for a triangle is equal to the base times the height -- now this so far, base times height is the area of a rectangle. So in order to get the area of the triangle, you're going to multiply that times 1/2. So 1/2 base times height. So in our example it's going to be 1/2 times 7 centimeters times 4 centimeters. We know what 7 times 4 is. We already know it's 28 centimeters -- we did that up there. So this right here is 28 centimeters. Then we want centimeters and we want to multiply that by 1/2. So that's going to be 14 centimeters just like that. So the area of this triangle is exactly 1/2 of the area of that rectangle. Now, the perimeter of this triangle becomes a little bit more complicated because figuring out this distance isn't the easiest thing in the world. Well, it will be easy for you once you get exposed to the Pythagorean Theorem. But I'm going to skip that right now. I'm going to leave that for the Pythagorean Theorem video. Let me just give you one more area of a triangle. Let's say I have a triangle that looks like this. This was a very special case that I drew to make it look like half of a rectangle. Let's say we had a triangle that looks like this. It's a little bit more skewed looking like this. And let's say that this distance down here is 3 meters -- that distance is 3 meters. Let's say we don't know what that distance is and we don't know what that distance is. But we do know that if we were to kind of drop a line straight down like this -- if you imagine this was a building or some type of mountain and you just drop something straight down onto the ground like that, we know that this distance is equal to -- let's say it's equal to 4 meters. So what is the area of this triangle going to be? Well, we apply the same formula. Area is equal to 1/2 base times height. So it's equal to 1/2 -- the base is literally this base right here of this triangle. So 1/2 times 3 times the height of the triangle. I guess a better way to think of it is an altitude of the triangle. So this thing isn't even in the triangle, but it is literally the height. If you imagine this was a building, you say how high is the building, it would be this height right there. So 1/2 times 3 times 4. You use that distance right there. Which is equal to 3 times 4 is 12 times 1/2 is equal to 6. We're going to be dealing with square meters. I really want to highlight the idea, because if I gave you a triangle that looked like this, where if this was 3 meters down here, and then if I were to tell you that this side over here is 4 meters, this is not something that you can just apply this formula to and figure out. In fact, you'd have to know some of the angles and whatnot to really be able to figure out the area, or you'd have to know this other side here. So this is not easy. You have to know what the altitude or the height of the triangle is. You need to know this distance. In this case, it was one of the sides, but in this case it's not one of the sides. You'd have to figure out what that side right there on the right hand side is in order to apply this formula.
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