I've got a square here.

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.