0:00:00.830,0:00:03.000
I've got a square here.

0:00:04.790,0:00:08.060
What makes it a square is[br]all of the sides are equal.

0:00:08.060,0:00:10.380
I haven't gone in depth into[br]angles yet, but these are at

0:00:10.380,0:00:12.520
right angles to each other.

0:00:12.520,0:00:13.470
I'll just draw it like that.

0:00:13.470,0:00:16.760
That means that if this bottom[br]side goes straight left and

0:00:16.760,0:00:19.880
right, that this left side[br]will go straight up and down.

0:00:19.880,0:00:22.210
That's all the right[br]angle really means.

0:00:22.210,0:00:27.290
Let's say that the side down[br]here is equal to 8 meters.

0:00:27.290,0:00:28.540
This side right here.

0:00:28.540,0:00:30.100
And this is a square.

0:00:30.100,0:00:35.980
And I were to ask you what[br]is the area of the square?

0:00:35.980,0:00:39.040
Well, the area is essentially[br]how much space the square

0:00:39.040,0:00:41.430
takes up, let's say, on[br]your screen right now.

0:00:41.430,0:00:46.040
So it's essentially a way of[br]measuring how much space

0:00:46.040,0:00:49.110
something takes up on kind of[br]a two-dimensional surface.

0:00:49.110,0:00:52.170
A two-dimensional surface would[br]just be this computer screen or

0:00:52.170,0:00:55.530
your piece of paper, if you're[br]also doing this problem.

0:00:55.530,0:00:58.680
An analogy would be if you had[br]an 8 meter by 8 meter room, how

0:00:58.680,0:01:01.570
much carpeting would you need[br]is kind of the size of the

0:01:01.570,0:01:04.240
space you need to fill out in[br]two dimensions on some

0:01:04.240,0:01:05.500
type of surface.

0:01:05.500,0:01:09.750
So the area here is literally[br]how much is this size that

0:01:09.750,0:01:11.980
you're filling up, and it's[br]very easy to figure

0:01:11.980,0:01:12.605
out for a square.

0:01:12.605,0:01:15.830
It's literally going to be your[br]base times your height -- and

0:01:15.830,0:01:18.570
this is true for any rectangle[br]-- but since this is a square,

0:01:18.570,0:01:20.650
your base and your height are[br]going to be the same number.

0:01:20.650,0:01:22.340
It's going to be 8 meters.

0:01:22.340,0:01:27.930
So your area is going to be 8[br]meters times 8 meters, which is

0:01:27.930,0:01:32.020
equal to 8 times 8 is 64, and[br]then your meters times your

0:01:32.020,0:01:34.580
meters -- you have to do the[br]same thing with your units --

0:01:34.580,0:01:37.200
you get 64 meters squared.

0:01:37.200,0:01:40.860
Or another way of saying,[br]this is 64 square meters.

0:01:40.860,0:01:44.390
You might be asking where[br]are those 64 square meters?

0:01:44.390,0:01:46.615
Well, you can actually[br]break it out here.

0:01:46.615,0:01:48.470
So let me draw it a[br]little bit bigger than

0:01:48.470,0:01:49.630
I originally drew it.

0:01:49.630,0:01:51.890
I probably should have drawn[br]it this big to begin with.

0:01:51.890,0:01:55.940
So let's say that's[br]my same square.

0:01:55.940,0:01:58.100
I'm going to draw a little[br]bit, so let me divide

0:01:58.100,0:02:00.240
it in the middle.

0:02:00.240,0:02:03.770
Let me see, I have -- and[br]we divide them again.

0:02:03.770,0:02:07.142
Then we divide each side[br]again just like that.

0:02:07.142,0:02:08.410
I could probably do it neater.

0:02:08.410,0:02:10.930
And let me do it one more time.

0:02:10.930,0:02:16.840
Divide these just like that,[br]and then divide these

0:02:16.840,0:02:19.010
just like that.

0:02:19.010,0:02:20.940
There you go.

0:02:20.940,0:02:21.480
OK.

0:02:21.480,0:02:23.980
Now the reason why I did this[br]is to show you the dimensions

0:02:23.980,0:02:27.030
along the base and the height.

0:02:27.030,0:02:30.650
We said this is 8 meters,[br]and notice I have 1, 2,

0:02:30.650,0:02:34.610
3, 4, 5, 6, 7, 8 meters.

0:02:34.610,0:02:36.620
And the same thing[br]along this side.

0:02:36.620,0:02:42.050
1, 2, 3, 4, 5, 6, 7, 8 meters.

0:02:42.050,0:02:45.340
So when we're talking about[br]64 square meters, we're

0:02:45.340,0:02:47.520
literally counting each[br]of the square meters.

0:02:47.520,0:02:50.380
A square meter is a[br]two-dimensional measurement,

0:02:50.380,0:02:51.780
that's 1 meter on each side.

0:02:51.780,0:02:53.490
That's 1 meter, that's 1 meter.

0:02:53.490,0:02:56.480
What I'm shading here in[br]yellow is 1 square meter.

0:02:56.480,0:02:59.030
And you could imagine just[br]counting the square meters.

0:02:59.030,0:03:05.070
In each row we're going to[br]have 1, 2, 3, 4, 5, 6,

0:03:05.070,0:03:07.080
7, 8 square meters.

0:03:07.080,0:03:08.610
And then we have 8 rows.

0:03:08.610,0:03:11.200
So we're going to have 8[br]times 8 square meters

0:03:11.200,0:03:12.760
or 64 meters square.

0:03:12.760,0:03:14.840
Which is essentially if you sat[br]here and just counted each of

0:03:14.840,0:03:19.050
these, you would count[br]64 square meters.

0:03:19.050,0:03:21.540
Now, what happens if I[br]were to ask you the

0:03:21.540,0:03:24.690
perimeter of my square?

0:03:28.000,0:03:30.620
The perimeter is the distance[br]you need to go to go

0:03:30.620,0:03:31.950
around the square.

0:03:31.950,0:03:33.990
It's not measuring,[br]for example, how much

0:03:33.990,0:03:35.070
carpeting you need.

0:03:35.070,0:03:37.520
It's measuring, for example, if[br]you wanted to put a fence

0:03:37.520,0:03:40.050
around your carpet -- I'm kind[br]of mixing the indoor and

0:03:40.050,0:03:42.400
outdoor analogies -- it would[br]be how much fencing

0:03:42.400,0:03:43.110
you would need.

0:03:43.110,0:03:46.210
So it would be the[br]distance around.

0:03:46.210,0:03:48.950
So it would be that distance[br]plus that distance plus that

0:03:48.950,0:03:50.980
distance plus that distance.

0:03:50.980,0:03:53.830
But we already know this[br]distance right here on the

0:03:53.830,0:03:58.020
bottom, we already know[br]this distance is 8 meters.

0:03:58.020,0:04:01.480
Then we know that the height[br]right here is 8 meters.

0:04:01.480,0:04:02.180
It's a square.

0:04:02.180,0:04:04.570
This distance up here is going[br]to be the same as this distance

0:04:04.570,0:04:07.710
down here -- it's going[br]to be another 8 meters.

0:04:07.710,0:04:09.450
Then when you go down the[br]left hand side it's going

0:04:09.450,0:04:11.380
to be another 8 meters.

0:04:11.380,0:04:15.670
We have four sides -- 1, 2, 3,[br]4 -- each of them are 8 meters.

0:04:15.670,0:04:18.660
So you add 8 to itself 4 times,[br]that's the same thing as 8

0:04:18.660,0:04:21.070
times 4, you get 36 meters.

0:04:21.070,0:04:25.050
Now notice, when we measured[br]just the amount of fencing we

0:04:25.050,0:04:28.530
needed, we ended up just with[br]meters, just with kind of a

0:04:28.530,0:04:30.680
one-dimensional measurement.

0:04:30.680,0:04:33.080
That's because we're not[br]measuring square meters here.

0:04:33.080,0:04:35.310
We're not measuring how[br]much area we're taking up.

0:04:35.310,0:04:38.560
We're measuring a distance[br]-- a distance to go around.

0:04:38.560,0:04:40.920
We are taking turns, but you[br]can imagine straightening out

0:04:40.920,0:04:44.570
this fence, and it would just[br]become one big fence like this,

0:04:44.570,0:04:48.160
which would have the same[br]length of 36 meters.

0:04:48.160,0:04:51.010
So that's why we just have[br]meters there for perimeter.

0:04:51.010,0:04:53.640
But for area we got square[br]meters, because we're counting

0:04:53.640,0:04:56.220
these two-dimensional[br]measurements.

0:04:56.220,0:04:58.840
Now, let's make it a little[br]bit more interesting.

0:04:58.840,0:05:02.070
What happens if instead[br]of a square I have a

0:05:02.070,0:05:05.780
rectangle like this?

0:05:09.700,0:05:15.280
Let's say that this side[br]over here is 7 centimeters.

0:05:15.280,0:05:23.170
And let's say that the height[br]right here is 4 centimeters.

0:05:23.170,0:05:25.845
So what is the area of this[br]rectangle going to be?

0:05:25.845,0:05:28.280
It's going to be 7[br]times 4 centimeters.

0:05:28.280,0:05:31.490
7 centimeters times[br]4 centimeters.

0:05:31.490,0:05:36.390
Remember, we could draw 7 rows,[br]right, and each of them is

0:05:36.390,0:05:39.540
going to have 4 square[br]centimeters -- each of those

0:05:39.540,0:05:40.380
is a square centimeter.

0:05:40.380,0:05:42.360
So if you were to count them[br]all out, you'd have 7 times

0:05:42.360,0:05:44.170
4 square centimeters.

0:05:44.170,0:05:45.140
It's 4 centimeters.

0:05:45.140,0:05:50.390
So it's equal to 28 centimeters[br]square or squared centimeters.

0:05:50.390,0:05:51.070
What's the perimeter?

0:05:55.260,0:05:58.660
Well, it's going to be equal to[br]this distance down here, which

0:05:58.660,0:06:03.670
is 7 centimeters, plus this[br]distance over here which is 4

0:06:03.670,0:06:07.480
centimeters, plus the distance[br]on the top -- this is a

0:06:07.480,0:06:09.170
rectangle, it's going to be[br]the same distance as

0:06:09.170,0:06:10.440
this one over here.

0:06:10.440,0:06:13.170
So plus another 7 centimeters.

0:06:13.170,0:06:16.300
Then you're going to have this[br]distance on the left hand side.

0:06:16.300,0:06:18.870
But this distance on the left[br]hand side is the same as this

0:06:18.870,0:06:21.810
distance right here -- this[br]is also 4 centimeters.

0:06:21.810,0:06:24.450
So plus another 4 centimeters.

0:06:24.450,0:06:25.450
And what do you get?

0:06:25.450,0:06:27.570
You get 7 plus 4 which is[br]11, and then you have

0:06:27.570,0:06:29.020
another 7 plus 4.

0:06:29.020,0:06:33.020
You have 11 plus 11, so[br]you have 22 centimeters.

0:06:33.020,0:06:36.300
Once again, it's not[br]a square centimeter.

0:06:36.300,0:06:42.300
Now let's divert -- let's go[br]away from our rectangle analogy

0:06:42.300,0:06:43.760
or our rectangle examples.

0:06:43.760,0:06:46.930
So let's see if we can do[br]the same with triangles.

0:06:46.930,0:06:49.940
So let's say I have[br]a triangle here.

0:06:49.940,0:06:52.100
I have a triangle like this.

0:06:54.990,0:06:58.720
Let's say that this distance[br]right here -- actually

0:06:58.720,0:06:59.760
let me draw it like this.

0:06:59.760,0:07:02.210
I think this'll make it a[br]little bit easier for you

0:07:02.210,0:07:04.550
to see how this relates[br]to a rectangle.

0:07:04.550,0:07:05.810
Let me draw it like this.

0:07:09.360,0:07:09.810
There you go.

0:07:09.810,0:07:11.300
That's my triangle.

0:07:11.300,0:07:14.510
And let's say that this[br]distance right here is 7

0:07:14.510,0:07:17.210
centimeters right down there.

0:07:17.210,0:07:21.090
And let's say that the[br]height of this triangle

0:07:21.090,0:07:23.520
is 4 centimeters.

0:07:23.520,0:07:26.160
And I were to ask you what is[br]the area of the triangle?

0:07:33.690,0:07:36.590
Well, when we had a rectangle[br]like this, we just

0:07:36.590,0:07:38.660
multiplied 7 times 4.

0:07:38.660,0:07:39.600
But what would that give us?

0:07:39.600,0:07:42.610
That would give us the area[br]of an entire rectangle.

0:07:42.610,0:07:44.610
If we did 7 times 4, that[br]would give us the area of

0:07:44.610,0:07:46.050
this entire rectangle.

0:07:46.050,0:07:49.640
You could imagine extending[br]my triangle up like this.

0:07:49.640,0:07:51.880
This is a right triangle --[br]this is going straight up and

0:07:51.880,0:07:54.420
down, this is going straight[br]left and right on the

0:07:54.420,0:07:55.910
bottom right here.

0:07:55.910,0:07:58.910
It's a 90 degree angle, if[br]you've been exposed to the

0:07:58.910,0:08:00.040
idea of angles already.

0:08:00.040,0:08:03.460
So you could almost view it as[br]it's 1/2 of this rectangle.

0:08:03.460,0:08:04.610
Not really almost, it is.

0:08:04.610,0:08:07.580
Because if you just double this[br]guy, you could imagine if you

0:08:07.580,0:08:12.190
flip this triangle over, you[br]get the same triangle but

0:08:12.190,0:08:14.910
it's just upside down[br]and flipped over.

0:08:14.910,0:08:17.650
So if you think about when you[br]multiply 7 times 4, you're

0:08:17.650,0:08:25.140
getting the area of this entire[br]rectangle, which we

0:08:25.140,0:08:26.800
just did up here.

0:08:26.800,0:08:30.210
But we want to know the[br]area of the triangle.

0:08:30.210,0:08:33.190
We want to know just[br]this area right here.

0:08:33.190,0:08:36.290
You can see, hopefully, from[br]this drawing that the area of

0:08:36.290,0:08:39.390
this triangle is exactly 1/2[br]of the area of the

0:08:39.390,0:08:40.990
entire rectangle.

0:08:40.990,0:08:47.040
So the area for a triangle is[br]equal to the base times the

0:08:47.040,0:08:50.490
height -- now this so far,[br]base times height is the

0:08:50.490,0:08:52.150
area of a rectangle.

0:08:52.150,0:08:53.755
So in order to get the area of[br]the triangle, you're going

0:08:53.755,0:08:55.910
to multiply that times 1/2.

0:08:55.910,0:08:58.160
So 1/2 base times height.

0:08:58.160,0:09:04.320
So in our example it's going[br]to be 1/2 times 7 centimeters

0:09:04.320,0:09:07.020
times 4 centimeters.

0:09:07.020,0:09:10.780
We know what 7 times 4 is.

0:09:10.780,0:09:13.880
We already know it's[br]28 centimeters -- we

0:09:13.880,0:09:15.710
did that up there.

0:09:15.710,0:09:19.050
So this right here[br]is 28 centimeters.

0:09:19.050,0:09:22.070
Then we want centimeters and we[br]want to multiply that by 1/2.

0:09:22.070,0:09:26.720
So that's going to be 14[br]centimeters just like that.

0:09:26.720,0:09:29.950
So the area of this triangle[br]is exactly 1/2 of the

0:09:29.950,0:09:31.700
area of that rectangle.

0:09:31.700,0:09:35.670
Now, the perimeter of this[br]triangle becomes a little bit

0:09:35.670,0:09:43.380
more complicated because[br]figuring out this distance

0:09:43.380,0:09:45.320
isn't the easiest[br]thing in the world.

0:09:45.320,0:09:47.965
Well, it will be easy for[br]you once you get exposed to

0:09:47.965,0:09:48.870
the Pythagorean Theorem.

0:09:48.870,0:09:50.290
But I'm going to skip[br]that right now.

0:09:50.290,0:09:54.010
I'm going to leave that for the[br]Pythagorean Theorem video.

0:09:54.010,0:09:58.450
Let me just give you one[br]more area of a triangle.

0:09:58.450,0:10:00.120
Let's say I have a triangle[br]that looks like this.

0:10:00.120,0:10:03.190
This was a very special case[br]that I drew to make it look

0:10:03.190,0:10:04.520
like half of a rectangle.

0:10:04.520,0:10:07.220
Let's say we had a triangle[br]that looks like this.

0:10:07.220,0:10:11.650
It's a little bit more[br]skewed looking like this.

0:10:11.650,0:10:19.346
And let's say that this[br]distance down here is 3 meters

0:10:19.346,0:10:21.950
-- that distance is 3 meters.

0:10:21.950,0:10:25.230
Let's say we don't know what[br]that distance is and we don't

0:10:25.230,0:10:26.570
know what that distance is.

0:10:26.570,0:10:30.660
But we do know that if we were[br]to kind of drop a line straight

0:10:30.660,0:10:32.670
down like this -- if you[br]imagine this was a building or

0:10:32.670,0:10:34.760
some type of mountain and you[br]just drop something straight

0:10:34.760,0:10:38.850
down onto the ground like that,[br]we know that this distance

0:10:38.850,0:10:43.770
is equal to -- let's say[br]it's equal to 4 meters.

0:10:43.770,0:10:46.140
So what is the area of this[br]triangle going to be?

0:10:50.420,0:10:52.910
Well, we apply the[br]same formula.

0:10:52.910,0:10:57.170
Area is equal to 1/2[br]base times height.

0:10:57.170,0:11:00.490
So it's equal to 1/2 -- the[br]base is literally this base

0:11:00.490,0:11:02.260
right here of this triangle.

0:11:02.260,0:11:07.380
So 1/2 times 3 times the[br]height of the triangle.

0:11:07.380,0:11:08.740
I guess a better way[br]to think of it is an

0:11:08.740,0:11:10.570
altitude of the triangle.

0:11:10.570,0:11:12.760
So this thing isn't even in[br]the triangle, but it is

0:11:12.760,0:11:13.820
literally the height.

0:11:13.820,0:11:15.850
If you imagine this was a[br]building, you say how high is

0:11:15.850,0:11:18.360
the building, it would be[br]this height right there.

0:11:18.360,0:11:20.395
So 1/2 times 3 times 4.

0:11:20.395,0:11:22.880
You use that distance[br]right there.

0:11:22.880,0:11:27.860
Which is equal to 3 times 4 is[br]12 times 1/2 is equal to 6.

0:11:27.860,0:11:30.830
We're going to be dealing[br]with square meters.

0:11:30.830,0:11:34.140
I really want to highlight the[br]idea, because if I gave you a

0:11:34.140,0:11:40.000
triangle that looked like this,[br]where if this was 3 meters down

0:11:40.000,0:11:44.250
here, and then if I were to[br]tell you that this side over

0:11:44.250,0:11:50.930
here is 4 meters, this is not[br]something that you can just

0:11:50.930,0:11:52.820
apply this formula[br]to and figure out.

0:11:52.820,0:11:54.790
In fact, you'd have to know[br]some of the angles and whatnot

0:11:54.790,0:11:56.840
to really be able to figure out[br]the area, or you'd have to

0:11:56.840,0:11:58.350
know this other side here.

0:11:58.350,0:12:02.480
So this is not easy.

0:12:02.480,0:12:05.890
You have to know what the[br]altitude or the height

0:12:05.890,0:12:06.720
of the triangle is.

0:12:06.720,0:12:07.900
You need to know this distance.

0:12:07.900,0:12:11.330
In this case, it was one of[br]the sides, but in this case

0:12:11.330,0:12:12.290
it's not one of the sides.

0:12:12.290,0:12:15.840
You'd have to figure out what[br]that side right there on the

0:12:15.840,0:12:19.590
right hand side is in order[br]to apply this formula.