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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
-
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
-
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
-
centimeters, plus the distance
on the top -- this is a
-
rectangle, it's going to be
the same distance as
-
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
-
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
-
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
-
centimeters right down there.
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And let's say that the
height of this triangle
-
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
-
bottom right here.
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It's a 90 degree angle, if
you've been exposed to the
-
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
-
entire rectangle.
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So the area for a triangle is
equal to the base times the
-
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
-
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
-
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
-
area of that rectangle.
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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.
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Well, it will be easy for
you once you get exposed to
-
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
-
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
-
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
-
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
-
altitude of the triangle.
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So this thing isn't even in
the triangle, but it is
-
literally the height.
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If you imagine this was a
building, you say how high is
-
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
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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
-
here is 4 meters, this is not
something that you can just
-
apply this formula
to and figure out.
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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.
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So this is not easy.
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You have to know what the
altitude or the height
-
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
-
right hand side is in order
to apply this formula.
