Lesson Video: Areas of Rectangles with the Same Perimeter Mathematics • 3rd Grade

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.