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# Perimeter & area

Perimeter is the distance around the outside of a shape. Area measures the space inside a shape. Learn how to calculate perimeter and area for various shapes. Created by Sal Khan.

Video transcript

What I want to do in this video
is a fairly straightforward primer on perimeter and area. And I'll do perimeter
here on the left, and I'll do area
here on the right. And you're probably pretty
familiar with these concepts, but we'll revisit it
just in case you are not. Perimeter is
essentially the distance to go around something
or if you were to put a fence around
something or if you were to measure-- if you were to
put a tape around a figure, how long that tape would be. So for example, let's
say I have a rectangle. And a rectangle is a figure that
has 4 sides and 4 right angles. So this is a
rectangle right here. I have 1, 2, 3, 4 right angles. And it has 4 sides,
and the opposite sides are equal in length. So that side is going to be
equal in length to that side, and that side is equal
in length to that side. And maybe I'll label the
points A, B, C, and D. And let's say we
know the following. And we know that
AB is equal to 7, and we know that
BC is equal to 5. And we want to know, what
is the perimeter of ABCD? So let me write it down. The perimeter of
rectangle ABCD is just going to be equal to the sum
of the lengths of the sides. If I were to build a fence, if
this was like a plot of land, I would just have
to measure-- how long is this side
right over here? Well, we already know
that's 7 in this color. So it's that side right
over there is of length 7. So it'll be 7 plus
this length over here, which is going to be 5. They tell us that. BC is 5. Plus 5. Plus DC is going to
be the same length is AB, which is
going to be 7 again. So plus 7. And then finally, DA a or AD,
however you want to call it, is going to be the same length
as BC, which is 5 again. So plus 5 again. So you have 7 plus 5 is 12
plus 7 plus 5 is 12 again. So you're going to
have a perimeter of 24. And you could go the
other way around. Let's say that you
have a square, which is a special case
of a rectangle. A square has 4 sides and 4 right
angles, and all of the sides are equal. So let me draw a square here. My best attempt. So this is A, B, C, D. And
we're going to tell ourselves that this right
here is a square. And let's say that this
square has a perimeter. So square has a perimeter of 36. So given that, what is the
length of each of the sides? Well, all the sides are going
to have the same length. Let's call them x. If AB is x, then BC is x,
then DC is x, and AD is x. All of the sides are congruent. All of these segments
are congruent. They all have the same
measure, and we call that x. So if we want to figure
out the perimeter here, it'll just be x plus x
plus x plus x, or 4x. Let me write that. x
plus x plus x plus x, which is equal to 4x, which
is going to be equal to 36. They gave us that
in the problem. And to solve this, 4
times something is 36, you could solve that
probably in your head. But we could divide
both sides by 4, and you get x is equal to 9. So this is a 9 by 9 square. This width is 9. This is 9, and then the height
right over here is also 9. So that is perimeter. Area is kind of a
measure of how much space does this thing take
up in two dimensions? And one way to think about area
is if I have a 1-by-1 square, so this is a 1-by-1 square--
and when I say 1-by-1, it means you only have
to specify two dimensions for a square or a rectangle
because the other two are going to be the same. So for example, you could
call this a 5 by 7 rectangle because that immediately
tells you, OK, this side is 5 and that side is 5. This side is 7,
and that side is 7. And for a square, you could
say it's a 1-by-1 square because that specifies
all of the sides. You could really say,
for a square, a square where on one side is 1,
then really all the sides are going to be 1. So this is a 1-by-1 square. And so you can view
the area of any figure as how many 1-by-1 squares
can you fit on that figure? So for example, if we were going
back to this rectangle right here, and I wanted to find out
the area of this rectangle-- and the notation
we can use for area is put something in brackets. So the area of rectangle
ABCD is equal to the number of 1-by-1 squares we can
fit on this rectangle. So let's try to do
that just manually. I think you already
might get a sense of how to do it a little bit quicker. But let's put a bunch of 1-by-1. So let's see. We have 5 1-by-1 squares
this way and 7 this way. So I'm going to try my
best to draw it neatly. So that's 1, 3, 3,
4, 5, 6, and then 7. 1, 2, 3, 4, 5, 6, 7. So going along one of the
sides, if we just go along one of the sides like
this, you could put 7 just along one side just like that. And then over here,
how many can we see? We see that's 1 row. And that's 2 rows. Then we have 3 rows and
then 4 rows and then 5 rows. 1, 2, 3, 4, 5. And that makes sense because
this is 1, 1, 1, 1, 1. Should add up to 5. These are 1, 1, 1, 1, 1, 1, 1. Should add up to 7. Yup, there's 7. So this is 5 by 7. And then you could
actually count these, and this is kind of straight
forward multiplication. If you want to know the
total number of cubes here, you could count it, or you can
say, well, I've got 5 rows, 7 columns. I'm going to have 35--
did I say cube-- squares. I have 5 squares in this
direction, 7 in this direction. So I'm going to have
35 total squares. So the area of this figure
right over here is 35. And so the general
method, you could just say, well, I'm just going to
take one of the dimensions and multiply it by
the other dimension. So if I have a
rectangle, let's say the rectangle is
1/2 by 1/2 by 2. Those are its dimensions. Well, you could
just multiply it. You say 1/2 times 2. The area here is going to be 1. And you might say, well,
what does 1/2 mean? Well, it means,
in this dimension, I could only fit 1/2
of a 1-by-1 square. So if I wanted to do
the whole 1-by-1 square, it's all distorted here. It would look like that. So I'm only doing half of one. I'm doing another half
of one just like that. And so when you add this
guy and this guy together, you are going to
get a whole one. Now what about area of a square? Well, a square is
just a special case where the length and
the width are the same. So if I have a square--
let me draw a square here. And let's call that XYZ-- I
don't know, let's make this S. And let's say I wanted
to find the area and let's say I know
one side over here is 2. So XS is equal to 2, and I
want to find the area of XYZS. So once again, I
use the brackets to specify the area of this
figure, of this polygon right here, this square. And we know it's a square. We know all the sides are equal. Well, it's a special
case of a rectangle where we would multiply the
length times the width. We know that they're
the same thing. If this is 2, then
this is going to be 2. So you just multiply 2 times 2. Or if you want to
think of it, you square it, which is
where the word comes from-- squaring something. So you multiply 2 times 2,
which is equal to 2 squared. That's where the
word comes from, finding the area of a
square, which is equal to 4. And you could see
that you could easily fit 4 1-by-1 squares
on this 2-by-2 square.