### Video Transcript

Areas of Rectangles with the Same
Perimeter

In this video, we’re going to learn
how to draw rectangles with the same perimeter. We’re going to count unit squares
to find their areas, and we’re going to compare these areas. This farmer wants to make a
rectangular enclosure for his pet rabbits to exercise in. But unfortunately, he’s a little
bit limited in what he can make because he only has 16 meters of fencing. And if he uses all the 16 meters
up, we can tell something about the rectangle. The perimeter of the enclosure,
that’s the distance all around it, is going to be 16 meters long. What size enclosure could he make
with this limited amount of fencing? Let’s try sketching an idea. Squared paper is always useful for
doing this.

Let’s imagine that the length of
each of the unit squares that we can see is worth one meter. Now, if our farmer makes an
enclosure with a width of one meter, this means he’s going to use two meters of his
16 meters altogether just on the ends of the enclosure. And he’s going to have 14 meters
left. So the two sides of the enclosure
are going to be worth seven meters. Two lots of seven meters plus two
lots of one meter equals 16 meters altogether. This doesn’t look like a great
enclosure for rabbits, does it really? There’s not a lot of space inside
it.

We know that the word we use to
describe the space inside the shape is its area, and we can measure the area of this
rectangle by counting the unit squares inside it. We can see that this particular
rectangle is made up of seven unit squares, so we can say its area is seven square
units. And because in this example we said
that each square length represents a meter, we can say the area is seven square
meters.

Now, for two rabbits who want to
run around and explore, this isn’t a lot of space, is it? The enclosure is quite long and
thin. Isn’t there anything else the
farmer could make using his 16 meters of fencing? Let’s see. What if he tries making a rectangle
this time with a width of two meters? This means that both ends of the
enclosure are going to be two meters long, which is four meters of his fencing
altogether he’s used. And because he only had 16 to start
with, he’s got 12 meters left. This means that the sides of the
enclosure are going to be six meters long. Two lots of six plus two lots of
two equals 16, so the perimeter is still 16 meters. Hopefully you can see this.

But the shape of the rectangle has
changed a little bit. It’s not quite so long anymore and
it is a little bit wider. Looks like there’s more space
inside there too. Let’s find out. If we count the unit squares inside
this rectangle, we can see that there are two rows of six. This means there are 12 square
units altogether. The area of the second enclosure is
going to be 12 square meters. So let’s stop and think about this
for a moment. The farmer has used exactly the
same amount of fencing, but the space inside the shape that he’s made has almost
doubled. Let’s show this using a table.

The first rectangle the farmer made
had a length of seven meters and a width of one meter. The perimeter of course was 16
meters. Seven plus seven plus one plus one
equals 16. And the area of this shape was
seven square meters. Then we tried a second rectangle
which had a length of six meters. This time, the width was two
meters, but again the perimeter was 16 meters because six add six add two add two
equals 16. And changing the dimensions of the
rectangle changed the area. This time, our area was 12 square
meters.

Now, by recording our results in a
table like this, there’s one or two things we could spot. Firstly, you might be able to see
what looks like a pattern between the length and the width each time. This would help us if we wanted to
try and find all the possibilities. We’ve tried a seven-by-one
rectangle then a six-by-two. Do you think maybe a five-by-three
rectangle adds up to 16-meter perimeter too? We’ll come back to think about this
at the end of the video. But for now, the main thing to
remember from this table is what we can see in the last two columns. Rectangles can have the same
perimeter but different areas. Just because the distance around
the outside is the same doesn’t mean the space inside is also the same.

Now, our farmer doesn’t look to
have solved his problem yet. Those rabbits are still looking a
little bit grumpy. Let’s leave the problem for a
moment and answer some questions where we practice what we’ve learned. Then we’ll come back and see if we
can help the farmer out.

Rectangle A and rectangle B have
the same perimeter. Which of them has a larger
area?

In the pictures underneath this
question, we can see two rectangles. These are labeled rectangle A and
rectangle B. Now, just by looking at these
rectangles, we can see that they’re slightly different sizes. But there’s something the same
about them. We’re told in the first sentence
that both rectangles have the same perimeter. Now we know that the perimeter of
the shape is the distance all around it. So this first sentence tells us
that the distance around both rectangles, although they look slightly different, is
the same.

The length of rectangle A is seven
centimeters. We can see this because it’s
labeled seven centimeters. But also if we count the squares,
we can see it’s seven squares long. Each square must be one centimeter
long. So rectangle A is made up of two of
these longer sides worth seven centimeters, and two lots of seven is 14. And then the width of this
rectangle is two centimeters and there are two sides that are worth this amount. So two lots of two equals four. And if we add 14 and four together,
we can see that the perimeter of rectangle A is 18 centimeters.

And if we quickly look at rectangle
B, we can see that the same is true. It has two sides with a length of
six centimeters. This is 12 centimeters
altogether. But the width of the rectangle is
three centimeters, so two sides are worth three centimeters, and three doubled is
six. So if we add all the sides
together, 12 and six equals 18 again. Rectangle A and rectangle B have
the same perimeter. But which of them has a larger
area? We know that the area of a shape is
the space inside it. And we can find the area of both of
these rectangles by counting the squares inside them.

We can see that rectangle A is made
up of two rows of seven squares. It has an area of 14 square
centimeters. In other words, 14 square
centimeters fit inside it. Now, if we look at rectangle B, we
can see it’s made up of three rows, and each row contains six squares. And three times six is 18. The area of this rectangle is 18
square centimeters. 18 square centimeters fit inside
it. And this tells us something really
interesting about shapes. They can have the same perimeter,
but they don’t have to have the same area. Both rectangles have the same
distance around them, but they don’t have the same space inside them. 18 square centimeters is greater
than 14 square centimeters, so the rectangle with a larger area is rectangle B.

Here is a rectangle. Select the rectangle that has the
same perimeter but a larger area than this one.

To begin with, in this question,
we’re given a picture of a rectangle. It’s labeled with a length of eight
centimeters and a width of four centimeters. And we know that the squares that
make up this rectangle must be one centimeter long because we can see that it’s
eight squares long and four squares wide. Now, underneath this we’re given
four more rectangles. And we’re told that we need to
select the rectangle that has the same perimeter but a larger area than our first
rectangle. Now we know that the perimeter of a
shape is the distance all around it.

So let’s find out what the
perimeter of our first rectangle is. As we’ve said already, its length
is eight centimeters, and its width is four centimeters. Eight and four is 12, so these two
sides together have a distance of 12 centimeters. But we need to double this amount
because we’ve got another side of eight centimeters and another side of four
centimeters. In other words, we can find the
perimeter by adding the length and the width together and then doubling it. Eight plus four is 12 and 12
doubled gives us a perimeter of 24 centimeters. So which of our possible answers
also has a perimeter of 24 centimeters?

The length of this first rectangle
is 11 centimeters, and its width is another one centimeter. So that’s 12 centimeters
altogether. But then, just like before, we’ve
got another length and another width, so we need to add together 11 and one and then
double it. And 12 doubled is 24. Although this rectangle looks a lot
different than our first one, it’s actually the same distance all the way
around. What about the second
rectangle? Its length is 10 centimeters; its
width is two centimeters. And 10 plus two is 12 again. And if we double 12, we get a
perimeter of 24. Do you notice that if we add
together the length and the width and they make 12, then we’re going to get a
perimeter of 24.

If we look at our next rectangle,
seven and five make 12. So this rectangle has the same
perimeter and so does the final rectangle. Nine plus three is 12. And if we had another nine and
another three, we get 24 centimeters. So in a way, this hasn’t really
helped us. We’ve still got four answers to
choose from. But at least it tells us something
about rectangles. They can have the same perimeter
but look very different. Now, we’re looking for a rectangle
that has the same perimeter but a larger area. And we know that the area of the
shape is the space inside it. And we can measure the space inside
these rectangles in square centimeters. We can do this just by counting the
squares inside them.

First of all, let’s find the area
of our first rectangle, the one we need to compare to. We can see that it’s made up of
four rows of squares and each row contains eight squares. And four times eight is a total of
32 square centimeters, so we’re looking for an area to beat this. The space inside our first
rectangle doesn’t look like it’s any bigger, does it? There are just 11 square
centimeters inside this shape. Our second rectangle is made up of
two rows, and each row contains 10 squares, so that’s 20 squares altogether. The area of this shape, 20 square
centimeters, is still less than 32.

Let’s keep looking. We can see inside our next
rectangle, there are five rows, and each row contains seven squares. And five times seven is a total of
35 square centimeters. This rectangle has more space
inside it than our first one. Looks like this might be the
correct answer. And if we look very quickly at our
final rectangle, we can see three rows of nine squares, which gives us an area of 27
square centimeters. This question shows us that even
though a rectangle may have the same perimeter as another one, they don’t always
have the same area. The rectangle that has the same
perimeter but a larger area than one with a length of eight centimeters and a width
of four centimeters is a rectangle that has a length of seven centimeters and a
width of five centimeters.

Now to end this video, let’s come
back to our farmer with his rabbit problem. If you remember, he only has 16
meters of fencing. And we started to draw a table to
help us find a rectangular shape that gave us the largest area for our rabbits to
play inside. Now, we could just keep trying to
find different rectangles until we find the one that gives the largest area. Or we could use what we’ve learned
in our last question to help us because the rectangle that gave us the largest area
wasn’t a long thin one. It was actually the rectangle that
was most like a square. Wasn’t a square, but it was almost
a square.

So let’s see whether we can use
this to help us. Let’s see whether he can make a
square or a rectangle that’s very close to being a square using his 16 meters of
fencing. He could make an enclosure with a
length of five meters and a width of three meters. Two lots of five is 10, two lots of
three is six, and 10 and six is a perimeter of 16 meters again. Or he could actually make a square
enclosure, where each side has a length of four meters. Now, which of these two remaining
rectangles do you think might have the larger area? Well, as we’ve seen already, long,
thin rectangles generally give us the smaller area. And the nearer a rectangle is to
being a square, that gives us the larger area.

Now the “squarest” of our farmer’s
enclosures is this one. And if we count the squares inside
it, we can see that there are four rows and there are four squares in each row. That’s an area of 16 square
meters. That’s the most yet. Let’s see if it gives us the
largest area though. To count the number of square
meters that there are inside our three-by-five rectangle, we can see that there are
three rows and there are five squares in each row. That’s a total of 15 square
meters. For the rabbits to have the most
space or area inside their enclosure, they’re going to need it to have a length of
four meters and a width of four meters because the “squarer” the rectangle, the
larger the area.

What have we learned in this
video? We’ve learned how to draw
rectangles with the same perimeter, how to count unit squares to find their areas,
and how to compare these areas. We’ve also learned that with
rectangles that have the same perimeter, the “squarer” the rectangle, the larger the
area.