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You can use your knowledge of perimeter and area to solve problems.

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Licence

This content is © Oak National Academy Limited (2023), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

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Key learning points

Problems in context involving perimeter can be solved.
Problems in context involving area can be solved.
The units of measurement should be the same.

Common misconception

Pupils may struggle to determine whether a question is asking about the perimeter or the area.

Encourage pupils to draw a diagram to represent the situation as this can make it clearer as to whether perimeter or area is required.

Keywords

Area - The area is the size of the surface and states the number of unit squares needed to completely cover that surface.
Perimeter - The perimeter is the distance around a 2D shape.

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Hi.

Welcome to today's lesson on problem solving with perimeter and area.

By the end of today's lesson, you'll have used your knowledge of perimeter and area to solve a variety of mathematical problems. Our lesson has three parts in it today.

We're going to start with part one which is called using one to find the other.

In other words, you're going to be given either the area or the perimeter and have to find the other one out.

Let's get started.

Knowledge of the perimeter of a shape could be useful when calculating the area.

Similarly, knowledge of the area of a shape could be useful when calculating the perimeter.

The perimeter of this shape is 115 centimetres.

What is the area of the shape? Hmm.

Well that's a triangle.

I know that to calculate the area I need to multiply the base and the perpendicular height and then halve it.

Do I already have the base perpendicular heights? No, I don't.

The 63 and the 28 are not perpendicular to each other.

Hmm.

I can see though that the missing length at the bottom is perpendicular to the length of 28, and in fact, if I had that, then I could work out the air of my triangle.

Does knowing the perimeter help me calculate that missing side length? You're right, it does.

Let's look at the working.

We know that 115 centimetres is my perimeter, I.

the total distance around my 2D shape.

That means that 115 is equal to 63, add 28, add in, whatever.

Now I'm gonna refer to this missing side as the base.

So the base of my triangle, well, 63 add 28 is 91.

So how much more do I need to get to 115? That's right.

I need 24, so the length of my base must be 24 centimetres.

Now I can use this information to calculate the area.

The area, remember of a triangle is the base times the perpendicular height divided by two.

Well, now I can put that information in.

That would be half of 24 multiplied by 28 or in other words, the area is 336.

The area of this shape is 225.

What's the perimeter going to be? Pause the video and work this one out.

Welcome back.

Did you start by using the fact that the area of a rectangle is the length times the width or in other words that 225 is equal to 25 times whatever the width of this shape is or 225 divided by 25 is nine, so therefore that missing width must be nine millimetres long.

Now that I know that, I can calculate the perimeter of the rectangle.

The perimeter would be 25 or 25 plus nine plus nine.

Now you may have written this more efficiently by saying it's two lots of 25, add two lots of nine or you could have written 25 add nine and then multiply that by two whichever working you've done, you'll get to the same answer that the perimeter of this rectangle is 68 millimetres.

Now a quick check.

You can see here there's a picture on the right and that is a composite rector linear shape.

What I wanna know is, is this statement true or false? I need to know the perimeter to work out the area for this shape.

So in other words, if I know the perimeter of this shape I'm then able to get enough information to calculate the area or is it false? In other words, you can already work out the area of this shape based on the information in the diagram and what you can deduce.

So is this true or false? And which of the justification supports your answer? A or B? Pause the video while you work this out now.

Welcome back.

Which one did you pick? It's false.

You can work out the area right now.

In fact, we could use method two from our lesson on calculating the area of composite recline shapes.

Method two, remember was to complete the rectangle so we could do 10 multiply by five and then subtract the area that we added in, which would be six multiplied by 2.

5 so you can actually work out the area right now.

You don't even need any of the other side lengths.

It's now time for our first task, we're going to be using perimeter to calculate area or area to calculate perimeter.

For A, I've told you the perimeter of the shape is 31 and I'd like you to calculate the area please.

Pause the video while you do this.

In part B, I've given you the area of this shape which is 63, and I'd like you to calculate the perimeter, pause the video while you do this.

Welcome back.

Let's go through these two.

For A, you were told the perimeter was 31.

This means that the perimeter, I.

the sum of all the edges is equal to 31.

While I can see that the two parallel slanted lengths will be six.

So I know the perimeter is six plus six plus two lots of the longer parallel side.

I'm referring to that as the base because I know I'm going to use this for my area calculation very soon.

So two lots of the base, well six had six is 12.

12 plus two lots of the base must be 31.

So two lots of the base must be 19 and therefore the base length is 9.

The area of a parallelogram is the base multiplied by the perpendicular height so 9.

5 times four given it an area of 38.

In B, you were told the area of this shape was 63 and I asked you to calculate the perimeter.

Remember, the area of the trapezium is the two parallel sides added together multiplied by the perpendicular distance between them and then halved.

Therefore, 63 is equal to half of the missing side length plus 13 multiplied by six or half of six is three and 63 divided by three is 21.

So 21 is equal to my missing side length added to 13 and that means that missing top length must be eight.

Perimeter is the distance around my shape.

So eight plus 13 and then the two slanted links are the same so in other words that's two lots of five, eight, add 13, add two, lots of five is 31, so the perimeter of my shape is 31 units.

It's now time for the second section of our lesson and this is where we are looking at perimeter and area problems within a context.

We did a little bit of this before in one of our previous lessons.

Do you remember Farmer Dylan and his sheep? We're going to look at some other perimeter and area context based questions now.

It's possible that you might need area or perimeter to solve a problem.

It is the context or the situation of a problem that can help us determine what we need to calculate.

Let's see what I mean by this.

We're going to sort these situations based on whether I'm asking you to calculate the area or the perimeter.

In other words, what makes more sense based on what it is you're trying to find out.

Put these different situations into either the perimeter box or the area box.

If you're working on paper, you can just put perimeter at the top of one list and area at the top of the other and just list underneath which one goes where.

Pause the video while you do this now.

Welcome back.

Let's see how you sorted them.

Fencing a garden and walking around a football pitch are definitely questions that are likely to be regarding perimeter.

Because if we are fencing a garden, the fence goes around the outside of the garden and likewise, walking around a football pitch means I'm looking at the distance around that football pitch.

The problems that involve area are likely to be the size of the field, painting a wall and tiling the bathroom because all three of these look at the size and possibly covering something rather than going around it.

Now the plot of land for a house is a debatable one.

If you put it in perimeter and justified it, then you are right.

If you put it in area and justified it then you are right because it really depends on whether you want to know the boundary, in other words going around the outside of the house or if you're thinking about the size of the land, which would be area.

So I'm happy if you put that in either as long as you had a good reason for it.

It's now time for our second task.

We've got a context here and you need to work out whether I'm asking you to calculate the perimeter or if I'm asking you to calculate the area.

When you've worked out which one I'm asking for then you'll need to solve the question.

In A, it says Aaron's garden is an unusual shape.

He wants to put a fence around it.

What is the total length of fencing he needs to buy? Pause the video while you have a go at this task.

Welcome back.

Let's now look at parts B and C.

Jun is a painter.

They want to create a mural.

How much space will this mural take up? And you can see a diagram of it below, hits a trapezium.

In C, I've told you that the rectangular wall Jun plans to paint on measures 21 by 38 centimetres.

Will the mural fit on the wall? You need to justify your answer.

Pause the video now while you have a go at parts B and C.

Welcome back.

It's now time to go through this.

Let's start with A, we need perimeter here.

Aaron is talking about length and the context is fencing which goes around our garden.

So we need the perimeter of our parallelogram.

That will be two lots of 10 and two lots of seven.

So the total length of fencing need to go around this garden is 20, add 14, which is 34 metres.

In B, we said Jun is a painter and they want to create a mural.

How much space will it take up? Well, that means we need area.

We want to know how much space is being covered.

The area of a trapezium is found by summing the two parallel sides, so that's 26 add 50, multiplying by the perpendicular distance between them.

That's 21 and then halving it.

When we work this calculation out, we get a total area of 798 square centimetres.

That's really quite a big painting.

Now let's look at part C.

The wall that Jun plans to paint on measures 21 by 38 centimetres.

Will the mural fit on the wall? Let's justify our answer.

Well, if we calculate the area of the wall, we see it is 21 times 38, 798 centimetre squared.

Well, this is the same area as the mural so we might think it will fit.

However, this is a practical question in context so let's think about that.

The wall measures 21 by 38 centimetres.

Remember that mural? In fact, let's go back and have a quick look at it.

See the distance at the bottom there? It's 50 centimetres long.

Wait a second, our wall measures 21 by 38.

If that mural has a base of 50 centimetres it's never going to fit on this wall.

It's not got a hope.

It's just simply too big.

In fact, try drawing it out on paper.

Now, I specifically talked about a painting and a tiny piece of a wall that's quite small I.

in centimetres because this way you can actually draw it on a piece of paper and see that it's never going to fit.

It's now time for part three of our lesson which is on units of measurement.

Now, we've been using a lots of different units of measurement throughout this unit on area and perimeter.

Let's see what we're talking about here.

In this example, area is found by calculating the number of unit squares needed to completely cover the solid blue cover.

But why do do we use squares and not a different shape? One of the properties of squares is that the lengths of all sides are equal.

This means that the units used to measure them should be the same.

So what we mean by that is in order for it to be a square, the two sides must be the same length.

Well then they have to use the same units or they wouldn't be the same length.

Now these shapes have not been drawn to scale.

I simply made them look this size so that they're easy and clear to see.

Are either of these shapes actually squares? How do you know? Well, let's consider the first one on the left.

I said that one length is 10 centimetres long and the other is one metre long.

Are these the same? Well, they're not.

One metre is the same as a hundred centimetres.

So actually I don't have a square here.

What I have is one length of 10 centimetres and the other of a hundred centimetres.

So if I drew my shape accurately we'd see it's very clearly a rectangle and not a square.

What's about here? I've got a length of two centimetres and a length of 20 millimetres.

Are these the same thing? Yes, they are.

20 millimetres is equivalent or the same as two centimetres.

Therefore these links are the same and I do have a square.

Now is this true or false? The shape you can see here is a square.

Is it true or false? And which of A or B justifies your answer? Pause the video now while you make your selection.

Welcome back.

Did you correctly say that it was true? 50 millimetres is equivalent to five centimetres and therefore this shape is indeed an accurately drawn square.

Side lengths should be converted so that they all use the same measurement before we carry out any calculations.

So for example, this square here, I should change one of those side lengths either to say that they are both five centimetres long and therefore the area is 25 square centimetres.

Or I should turn them both into millimetres and say that the area is 2,500 square millimetres.

Jacob and Lucas have both tried to calculate the area of this triangle.

Jacob thinks the area is 226.

6 square centimetres and Lucas thinks the area is 4.

536 square metres.

Each of them has made an error, however.

Can you help them? Pause the video and work out the error that each of them has made.

Welcome back.

Let's see how you got on.

What has Jacob done? Jacob's worked out 6.

3 times 72 and then halved it to get 226.

The problem is Jacob can't do that.

He should have converted the 6.

3 metres into centimetres first.

What about Lucas? Lucas worked out 6.

3 times 0.

72 is 4.

536.

So Lucas converted the 72 centimetres into metres.

Well done, Lucas, except he's finding the area of a triangle.

So what does he failed to do? He's not multiplied by a half.

He's found the area of a rectangle, not a triangle.

Sorry, Lucas.

It's now time for your very final task/ in it, Mariam wants to tile a wall in her bathroom.

The wall is rectangular and measures three metres by 4.

5 metres.

Each tile is a square which measures 50 centimetres by 50 centimetres.

How many tiles will Merriam need to cover the wall? Pause the video while you have a go at this task.

Welcome back.

Let's see what you said.

One tile has an area of 50 by 50.

The wall has an area of 300 by 450.

So what I did was I turned the measurements of the wall from metres into centimetres.

I then divided to say, well how many tiles are needed to cover that area? That'll be 54 tiles.

Did you see that a check your answer appeared on the screen? That's just one way of working this out.

But there are other ways.

We should try a different method just to make sure we get the same answer.

This is an alternative way of doing it.

Let's check our answer.

Now, you'll see here that I've got a diagram to represent the wall that I'm tiling.

It's 4.

5 metres long by three metres long.

Now, I know that a square tile measures 50 centimetres by 50 centimetres, or in other words half a metre by half a metre.

Well, that means that I need six tiles to cover down the side to make the length of three metres and nine tiles across the top to make the 4.

5 metres.

Well then I can just check by looking at my diagram.

I've got here six rows, each of nine tiles, well nine lots of six of 54, which is the same answer I got before.

Whew.

I now feel pretty confident that I'm right but it was good to check and it's a very different way of checking.

It involved drawing a diagram.

In math, it can make problems so much easier to draw a diagram, so don't hesitate to do this.

Let's summarise our learning for today.

Perimeter can be useful in calculating area.

An area can be useful in calculating perimeter.

The context of a problem can help you determine what you need to calculate, and any side lengths should be converted so they all use the same measurement before we carry out any calculations.

Well done.

You've worked really well today and this brings us to the end of our unit on area and perimeter.

I look forward to seeing you for our next unit of learning.

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. What is the perimeter?" }, "questionType": "Short answer", "correct": [ { "text": "48" }, { "text": "48 cm" } ], "id": "cfe5c72d-ad78-6874-f7da-9a29c6dcade1", "lessonId": "6a6a1745-7cf4-5021-9d82-b9c00db9e289" }, { "type": "Question", "question": { "text": "A groundskeeper walks around the boarder of the park he looks after. How far does he walk?", "image": "https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1690273957%2Frelascw8jgfb4qve2blw.png&w=640&q=75", "imageKey": "9cbbc439-e63d-d7a2-aec6-0cc8b13738f4/question/1ef40c021f3281a0b6c4797a56297395.png" }, "questionType": "Short answer", "correct": [ { "text": "460 m" }, { "text": "460" } ], "id": "f6b7f6c9-3857-d099-cfda-8b946deb615b", "lessonId": "6a6a1745-7cf4-5021-9d82-b9c00db9e289" }, { "type": "Question", "question": { "text": "Andeep is helping his family to lay a new carpet. How much carpet is needed to cover the floor of the room? (You do not need to state the units.)", "image": "https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1690273957%2Fsnjlbseeu0oke0sil5us.png&w=640&q=75", "imageKey": "9cbbc439-e63d-d7a2-aec6-0cc8b13738f4/question/8fa61d5c9791d221484e9ccda04c7793.png" }, "questionType": "Short answer", "correct": [ { "text": "11.25" }, { "text": "11.25 m

^{2}

" } ], "id": "1046a8d5-6e33-5e60-55d8-a6d154ac3d9f", "lessonId": "6a6a1745-7cf4-5021-9d82-b9c00db9e289" }, { "type": "Question", "question": { "text": "Calculate the perimeter of the rectangle.", "image": "https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1690273956%2Ftt9nmxkarlng1k97ir07.png&w=640&q=75", "imageKey": "9cbbc439-e63d-d7a2-aec6-0cc8b13738f4/question/05a7bcf27760cf6e9143df517828d2fb.png" }, "questionType": "Multiple choice", "correct": [ { "text": "636 cm" } ], "choices": [ { "text": "36.6 m" }, { "text": "318 cm" }, { "text": "339 m" }, { "text": "636 cm" } ], "id": "4fc3f2ef-5fb7-584a-b604-c9e133d723f2", "lessonId": "6a6a1745-7cf4-5021-9d82-b9c00db9e289" }, { "type": "Question", "question": { "text": "Calculate the area of the triangle (you do not need to state the units).", "image": "https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1690273957%2Fw2x6adfpk6qtdyz1egoa.png&w=1080&q=75", "imageKey": "9cbbc439-e63d-d7a2-aec6-0cc8b13738f4/question/91f58d83f906f3ca23e9e55dcb3d33ce.png" }, "questionType": "Short answer", "correct": [ { "text": "326 000" }, { "text": "326000" }, { "text": "32.6" } ], "id": "dd11cce1-9195-e4d1-fc17-f5bb7bbcb67b", "lessonId": "6a6a1745-7cf4-5021-9d82-b9c00db9e289" } ], "lessonId": "6a6a1745-7cf4-5021-9d82-b9c00db9e289", "meta": { "gradeStageId": "ks3", "subject": "Maths", "grade": "Year 7", "topic": null, "unit": "Perimeter and area", "lessonCount": "12", "age": [ 11, 12 ], "gradeId": "age_11_12", "gradeStage": "Key Stage 3", "lesson": "Problem solving with perimeter and area" } } ] }