WEBVTT

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I've got a square here.

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What makes it a square is
all of the sides are equal.

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I haven't gone in depth into
angles yet, but these are at

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right angles to each other.

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I'll just draw it like that.

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That means that if this bottom
side goes straight left and

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right, that this left side
will go straight up and down.

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That's all the right
angle really means.

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Let's say that the side down
here is equal to 8 meters.

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This side right here.

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And this is a square.

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And I were to ask you what
is the area of the square?

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Well, the area is essentially
how much space the square

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takes up, let's say, on
your screen right now.

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So it's essentially a way of
measuring how much space

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something takes up on kind of
a two-dimensional surface.

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A two-dimensional surface would
just be this computer screen or

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your piece of paper, if you're
also doing this problem.

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An analogy would be if you had
an 8 meter by 8 meter room, how

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much carpeting would you need
is kind of the size of the

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space you need to fill out in
two dimensions on some

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type of surface.

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So the area here is literally
how much is this size that

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you're filling up, and it's
very easy to figure

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out for a square.

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It's literally going to be your
base times your height -- and

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this is true for any rectangle
-- but since this is a square,

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your base and your height are
going to be the same number.

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It's going to be 8 meters.

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So your area is going to be 8
meters times 8 meters, which is

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equal to 8 times 8 is 64, and
then your meters times your

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meters -- you have to do the
same thing with your units --

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you get 64 meters squared.

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Or another way of saying,
this is 64 square meters.

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You might be asking where
are those 64 square meters?

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Well, you can actually
break it out here.

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So let me draw it a
little bit bigger than

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I originally drew it.

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I probably should have drawn
it this big to begin with.

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So let's say that's
my same square.

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I'm going to draw a little
bit, so let me divide

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it in the middle.

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Let me see, I have -- and
we divide them again.

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Then we divide each side
again just like that.

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I could probably do it neater.

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And let me do it one more time.

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Divide these just like that,
and then divide these

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just like that.

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There you go.

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OK.

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Now the reason why I did this
is to show you the dimensions

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along the base and the height.

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We said this is 8 meters,
and notice I have 1, 2,

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3, 4, 5, 6, 7, 8 meters.

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And the same thing
along this side.

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1, 2, 3, 4, 5, 6, 7, 8 meters.

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So when we're talking about
64 square meters, we're

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literally counting each
of the square meters.

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A square meter is a
two-dimensional measurement,

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that's 1 meter on each side.

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That's 1 meter, that's 1 meter.

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What I'm shading here in
yellow is 1 square meter.

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And you could imagine just
counting the square meters.

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In each row we're going to
have 1, 2, 3, 4, 5, 6,

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7, 8 square meters.

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And then we have 8 rows.

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So we're going to have 8
times 8 square meters

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or 64 meters square.

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Which is essentially if you sat
here and just counted each of

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these, you would count
64 square meters.

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Now, what happens if I
were to ask you the

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perimeter of my square?

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The perimeter is the distance
you need to go to go

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around the square.

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It's not measuring,
for example, how much

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carpeting you need.

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It's measuring, for example, if
you wanted to put a fence

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around your carpet -- I'm kind
of mixing the indoor and

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outdoor analogies -- it would
be how much fencing

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you would need.

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So it would be the
distance around.

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So it would be that distance
plus that distance plus that

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distance plus that distance.

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But we already know this
distance right here on the

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bottom, we already know
this distance is 8 meters.

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Then we know that the height
right here is 8 meters.

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It's a square.

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This distance up here is going
to be the same as this distance

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down here -- it's going
to be another 8 meters.

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Then when you go down the
left hand side it's going

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to be another 8 meters.

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We have four sides -- 1, 2, 3,
4 -- each of them are 8 meters.

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So you add 8 to itself 4 times,
that's the same thing as 8

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times 4, you get 36 meters.

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Now notice, when we measured
just the amount of fencing we

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needed, we ended up just with
meters, just with kind of a

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one-dimensional measurement.

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That's because we're not
measuring square meters here.

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We're not measuring how
much area we're taking up.

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We're measuring a distance
-- a distance to go around.

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We are taking turns, but you
can imagine straightening out

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this fence, and it would just
become one big fence like this,

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which would have the same
length of 36 meters.

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So that's why we just have
meters there for perimeter.

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But for area we got square
meters, because we're counting

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these two-dimensional
measurements.

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Now, let's make it a little
bit more interesting.

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What happens if instead
of a square I have a

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rectangle like this?

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Let's say that this side
over here is 7 centimeters.

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And let's say that the height
right here is 4 centimeters.

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So what is the area of this
rectangle going to be?

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It's going to be 7
times 4 centimeters.

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7 centimeters times
4 centimeters.

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Remember, we could draw 7 rows,
right, and each of them is

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going to have 4 square
centimeters -- each of those

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is a square centimeter.

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So if you were to count them
all out, you'd have 7 times

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4 square centimeters.

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It's 4 centimeters.

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So it's equal to 28 centimeters
square or squared centimeters.

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What's the perimeter?

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Well, it's going to be equal to
this distance down here, which

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is 7 centimeters, plus this
distance over here which is 4

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centimeters, plus the distance
on the top -- this is a

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rectangle, it's going to be
the same distance as

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this one over here.

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So plus another 7 centimeters.

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Then you're going to have this
distance on the left hand side.

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But this distance on the left
hand side is the same as this

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distance right here -- this
is also 4 centimeters.

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So plus another 4 centimeters.

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And what do you get?

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You get 7 plus 4 which is
11, and then you have

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another 7 plus 4.

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You have 11 plus 11, so
you have 22 centimeters.

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Once again, it's not
a square centimeter.

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Now let's divert -- let's go
away from our rectangle analogy

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or our rectangle examples.

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So let's see if we can do
the same with triangles.

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So let's say I have
a triangle here.

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I have a triangle like this.

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Let's say that this distance
right here -- actually

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let me draw it like this.

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I think this'll make it a
little bit easier for you

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to see how this relates
to a rectangle.

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Let me draw it like this.

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There you go.

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That's my triangle.

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And let's say that this
distance right here is 7

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centimeters right down there.

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And let's say that the
height of this triangle

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is 4 centimeters.

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And I were to ask you what is
the area of the triangle?

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Well, when we had a rectangle
like this, we just

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multiplied 7 times 4.

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But what would that give us?

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That would give us the area
of an entire rectangle.

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If we did 7 times 4, that
would give us the area of

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this entire rectangle.

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You could imagine extending
my triangle up like this.

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This is a right triangle --
this is going straight up and

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down, this is going straight
left and right on the

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bottom right here.

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It's a 90 degree angle, if
you've been exposed to the

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idea of angles already.

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So you could almost view it as
it's 1/2 of this rectangle.

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Not really almost, it is.

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Because if you just double this
guy, you could imagine if you

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flip this triangle over, you
get the same triangle but

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it's just upside down
and flipped over.

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So if you think about when you
multiply 7 times 4, you're

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getting the area of this entire
rectangle, which we

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just did up here.

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But we want to know the
area of the triangle.

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We want to know just
this area right here.

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You can see, hopefully, from
this drawing that the area of

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this triangle is exactly 1/2
of the area of the

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entire rectangle.

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So the area for a triangle is
equal to the base times the

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height -- now this so far,
base times height is the

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area of a rectangle.

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So in order to get the area of
the triangle, you're going

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to multiply that times 1/2.

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So 1/2 base times height.

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So in our example it's going
to be 1/2 times 7 centimeters

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times 4 centimeters.

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We know what 7 times 4 is.

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We already know it's
28 centimeters -- we

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did that up there.

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So this right here
is 28 centimeters.

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Then we want centimeters and we
want to multiply that by 1/2.

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So that's going to be 14
centimeters just like that.

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So the area of this triangle
is exactly 1/2 of the

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area of that rectangle.

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Now, the perimeter of this
triangle becomes a little bit

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more complicated because
figuring out this distance

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isn't the easiest
thing in the world.

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Well, it will be easy for
you once you get exposed to

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the Pythagorean Theorem.

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But I'm going to skip
that right now.

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I'm going to leave that for the
Pythagorean Theorem video.

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Let me just give you one
more area of a triangle.

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Let's say I have a triangle
that looks like this.

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This was a very special case
that I drew to make it look

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like half of a rectangle.

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Let's say we had a triangle
that looks like this.

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It's a little bit more
skewed looking like this.

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And let's say that this
distance down here is 3 meters

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-- that distance is 3 meters.

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Let's say we don't know what
that distance is and we don't

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know what that distance is.

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But we do know that if we were
to kind of drop a line straight

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down like this -- if you
imagine this was a building or

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some type of mountain and you
just drop something straight

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down onto the ground like that,
we know that this distance

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is equal to -- let's say
it's equal to 4 meters.

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So what is the area of this
triangle going to be?

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Well, we apply the
same formula.

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Area is equal to 1/2
base times height.

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So it's equal to 1/2 -- the
base is literally this base

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right here of this triangle.

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So 1/2 times 3 times the
height of the triangle.

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I guess a better way
to think of it is an

00:11:08.740 --> 00:11:10.570
altitude of the triangle.

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So this thing isn't even in
the triangle, but it is

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literally the height.

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If you imagine this was a
building, you say how high is

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the building, it would be
this height right there.

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So 1/2 times 3 times 4.

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You use that distance
right there.

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Which is equal to 3 times 4 is
12 times 1/2 is equal to 6.

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We're going to be dealing
with square meters.

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I really want to highlight the
idea, because if I gave you a

00:11:34.140 --> 00:11:40.000
triangle that looked like this,
where if this was 3 meters down

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here, and then if I were to
tell you that this side over

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here is 4 meters, this is not
something that you can just

00:11:50.930 --> 00:11:52.820
apply this formula
to and figure out.

00:11:52.820 --> 00:11:54.790
In fact, you'd have to know
some of the angles and whatnot

00:11:54.790 --> 00:11:56.840
to really be able to figure out
the area, or you'd have to

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know this other side here.

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So this is not easy.

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You have to know what the
altitude or the height

00:12:05.890 --> 00:12:06.720
of the triangle is.

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You need to know this distance.

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In this case, it was one of
the sides, but in this case

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it's not one of the sides.

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You'd have to figure out what
that side right there on the

00:12:15.840 --> 00:12:19.590
right hand side is in order
to apply this formula.