Area and Perimeter - Surface Areas & Volumes Video Lecture - Class 9 | Best Video for Class 9

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.