Lesson video

Hi, and welcome to my lesson on perimeter with composite rectilinear shapes.

By the end of this lesson you'll be able to use the properties of a range of polygons to deduce the perimeters of compound shapes including generalising a perimeter formula where it's appropriate.

The words that we're gonna be using today that might be unfamiliar to you are compound shape, composite shape, and composite rectilinear shape.

Now you might remember some of these before 'cause you will have encountered some of them in primary but it's okay if you've forgotten, it has been a while.

Pause the video and write down what you think each set of words might mean.

A compound shape is a shape created using two or more basic shapes.

A composite shape is an alternative term.

In other words, it means the same thing as a compound shape.

So you could say compound shape or composite shape.

Same thing.

Now a composite or compound rectilinear shape is one that is made only from rectangles.

So remember how we said that a composite or compound shape was made from basic shapes? Well, it could be lots of different basic shapes, but a composite rectilinear shape must be made from rectangles, no other shapes sneaking in there.

Now our lesson today has three parts and we're gonna start now with the first section called Strange Staircases.

Understanding perimeter means being able to reason with perimeter.

Identifying what is the same and what is different is an important skill for problem solving and we're gonna utilise that skill right now.

In this task you are told the perimeter of each rectangle is 26 centimetres.

How long is the red line? So how long is the red line for A, how long is the red line for B, and how long is the red line for C? You need to justify your answer.

Pause the video now and have a go.

Each line is 13 centimetres long.

That's right.

It doesn't matter if you're looking at A, B or C, that red line has the same total length each time.

That might seem really odd.

Of course, if you've justified it correctly you'll see that your reasoning is valid for all of them.

Let's just see how this works if you're a little uncertain.

We'll start by looking at A.

What would happen if all the vertical red lines moved to the right of the rectangle and all of the horizontal lines move straight down to the bottom of the rectangle? Where would those lines now be? What part of the rectangle are they touching and what would it look like? Here's an animation to help.

As you can see, the vertical lines are moving to the right and the horizontal lines are moving down.

When these red lines have moved, can you see that they cover one length and one width of our rectangle? In other words, they cover half of the perimeter of our rectangle.

If the perimeter is 26, then half of that must be 13.

By moving the red vertical lines all the way to the right and the red horizontal lines all the way down, you actually cover one length and one width of our rectangle.

Remember, the length and the width sum together, it's half of the perimeter of the entire rectangle.

Since our perimeter is 26, that meant the length of the red line must be half of that, or 13.

Why does this approach of moving the lines also work for B and C? Remember, we had to justify why that length of the red line in B and C was still the same, was still 13.

Well, those red lines are also only made of vertical and horizontal lines and they don't overlap.

So if we did the same process and we moved all the vertical ones to the right and all the horizontal ones down, we'd again perfectly cover one length and one width of our rectangle.

Now, some composite rectilinear shapes can also be a rearranged like this to form a rectangle.

Rearranging the signs this way is not going to change the perimeter of our shape.

Here's an example of a composite rectilinear shape.

So we can see that on the left.

If I move that horizontal red line down and the vertical red line across to the right you can see it turns into a rectangle.

Now that's interesting because I can now see that the two red lines that appear in the rectangle on the right are going to be equal to the two black lines.

In other words, rearranging those sides didn't change my perimeter.

All it did was just turn it into a more familiar looking shape.

Let's see if we can apply that knowledge to calculate the perimeter of the shape you see here.

Pause the video and work out which of the three measurements given is the correct perimeter of this shape.

Welcome back.

Did you choose 30 metres? If you did, well done.

We know that we can see that the width of our shape is 5 metres.

In other words, the total vertical distance is 5.

If we look across, we can see two other vertical sides and actually if I moved the top vertical side across so it sat above the 2.

6 I'd have a length that would also be 5.

I can see that the bottom distance is 10 and there are two horizontal distances above.

Again, if I place them next to each other the total will be 10.

So I've actually got here two lots of five and two lots of 10, giving me a total of 30.

Now if you're not comfortable with that method you could have worked it out a different way.

You could have said, I know that the total there is 5, and I've already got a length of 2.

6.

How much more do I need to make 5? And you could have worked out that that missing height was 2.

4.

Could have done similar reasoning to work out that the missing horizontal distance in my shape is 4.

3, and then you could have summed all the sides.

So there are other ways to make this work too.

Let's now look at section two, where we're practising some more with composite rectilinear shapes.

Now some composite rectilinear shapes can be easily rearranged to form a rectangle and we saw that in our first section, but this is not always the easiest method.

In this task, I'd like you to have a go at rearranging the sides of this composite rectilinear shape to form a rectangle.

How easy is it to do this? Write down what you found when you were having a go.

Pause the video now while you have a go at this task.

Welcome back.

How did you get on? Was it easy to move the sides of this composite rectilinear shape to make a rectangle? It isn't as easy as it was before, is it? It's not as straight to just move it left, right, up, or down.

Does this animation help? I started with a rectangle, and I'm showing where the sides would move in order to get back to that composite rectilinear shape that you started with.

So you can see there was a lot of rearranging involved.

So it is possible to rearrange this shape to form a rectangle, but it was not easy, because the sides to move currently overlap each other so I couldn't move them straight, left or right.

We're talking about those vertical sides of length three.

Instead I had to place them on top of the outside vertical sides and then move all the horizontal lines up.

Far more tricky.

In this particular instance I don't think it's easier to try and rearrange, what did you think? So for some shapes it may be much easier to sum the sides.

Here's our shape from our task and what I've done is I've worked out what the missing side lengths are.

I knew that the side that was opposite my length of 3 was also going to be 3, because those sides were marked with notation to show they're the same length.

I know that the side opposite the 6 right at the other end, so on the right hand side of my shape had to also be 6 because it started at the same point and went to the same point.

The horizontal line was a little bit tricky to work out.

I know the total distance across my shape is 8, and I can look up and see I've got two horizontal lengths, one of length 2, and one of length 4.

Well that gives me a total of 6.

How much more do I need to get to 8? I need 2 more.

That must be the length of that missing side.

Once I'd worked out those missing measurements I was then able to calculate the perimeter by summing all of those sides.

And as you can see, it was a long calculation.

Could you have done that calculation more efficiently? Think back to earlier lessons in this unit when we looked at how to efficiently calculate perimeter.

Maybe you'd like to pause the video and have a go at writing this more efficiently.

Which method would you look to use to find the perimeter of this shape? Pause the video and write down whether you'd prefer to rearrange this shape into a rectangle or whether you'd want to sum all of the sides.

State your reasoning to justify your answer.

Pause the video now to do this.

Welcome back.

Which method did you go for? You might have said something like this.

I would want to sum all the sides because this shape is not simple to rearrange, and if you did, I completely agree with you.

I would not wanna turn that into a rectangle.

It looks far too complicated.

Now let's move on and look at our final section today.

And this is where we'll be doing some perimeter calculations with composite rectilinear shapes.

We're going to calculate the perimeter of this composite rectilinear shape.

Now I've got some missing length here.

Is it possible for me to work out all of the missing lengths? That's right, it's not.

I can work out what the missing vertical side is because I can add 7.

4 to 2.

8 to calculate it.

The problem comes when I look at the missing horizontal lengths.

I've got two of them but I have no idea how long each one is.

What I do know, however, is what their total length will be.

What I've actually got here is two lots of the total height of my shape so two lots of what happens when I add 7.

4 to 2.

8, and I've also got two lots of that horizontal distance, the 4.

3.

So actually if I add those lengths together and double, I'll get the perimeter of my shape, which is 29 centimetres.

It's now your turn.

Calculate the perimeter of this composite rectilinear shape.

Pause the video while you do this.

How did you get on? Did you spot that at each side of the rectangle I have a total distance or a total length of 3 inches, and that if I look across, so horizontally, I have a total distance of 4.

8 both at the top and the bottom of my rectangle.

So what I've actually got is two lots of 4.

8 plus 3, or I could have said two lots of 4.

8 and two lots of 3, making a total of 15.

6 inches.

It's now time for our final task.

Here we have two composite rectilinear shapes.

For each shape, I'd like you to calculate the perimeter.

Think carefully about what lengths you know and what lengths you need to work out.

You don't always need to know every single individual length.

Sometimes just knowing what they add up to is good enough.

Keep that in mind as you have a go at this task.

Pause the video now while you complete it.

Welcome back.

Let's see how you got on as we go through our solutions.

Let's start with A.

The perimeter of my shape is two lots of the measurements I can see.

Let's be crystal clear about that though and make sure we understand why it's two lots of the measurements we can see.

You can see that I have a length there of 11.

1.

The two sides that are parallel to the 11.

1 that are unmarked and we don't know, we dunno what individual lengths are but we do know that the total of those two must be 11.

1.

In other words, 11.

1 plus 11.

1, or two lots of 11.

1.

Now let's look at the sides that are perpendicular to that.

The shortest length is measured is 0.

6 and the slightly longer one at 3.

2.

We know that if we add those two lengths together we'll get the total length of the top line.

In other words, that's 0.

6 add 3.

2, and we need to double it which is how we get to our working that we can see on the screen now.

When we work that calculation out we get that the total perimeter is 29.

8 metres.

Let's now look at B.

B was a far more complicated shape.

There was quite a lot going on here.

We can see that the total height of our shape is 23.

This means that all the vertical lines that are towards the left of the shape sum to 23 and all the vertical lines to the right of the shape sum to 23.

In total therefore, when I'm adding all the vertical lines together I can write this more efficiently as two lots of 23.

Now, let's consider the horizontal lines, and this is where things get a little trickier.

I can see that I have a distance of 17 at the top, I have a distance of 2 marked on, a distance of 5, a distance of 19 and another horizontal distance of 2.

Okay, so at the moment I can sum those together, but what I've got is that missing horizontal distance shown.

Hmm, I need to work out what that's going to be.

The distance across the top is 17.

I can then see underneath there's a horizontal distance of 2, and that's like me coming back on myself.

Effectively at that point, I've therefore only travelled a distance of 15 from where I started.

If I look at the next horizontal line it goes out to the right by 5, and if I'm at 15 and I add on 5, that's a total of 20.

So actually from the very left of my shape to the very right of my shape, it's a total length of 20.

If I use that knowledge to look at the distance of 19 then what I can see is that that extra little gap there must be a gap of 1.

Hang on a second.

I can see that just above, I come back on myself by 2, so to go from that point, which would therefore be 17 from the right, to go out to 20, it must be three.

That means that the missing length, that missing horizontal length must have a distance of 3.

Now I know that, I can work out my calculation, and get a total of 94 units.

You may have found this a lot easier if you drew a copy of this shape in your books or on some paper, and then had a go at writing down the sides that you were missing.

If you did, you may have found that doing those calculations made things just that little bit easier.

If you were lucky and you had a printed copy of this question, you could have just written straight onto it which you may have found a lot easier.

Thing to remember about composite rectilinear shapes is that some measurements can be deduced just by looking at other sides of our composite rectilinear shape.

Let's summarise our learning today.

Some composite rectilinear shapes can be easily rearranged to form a rectangle.

Rearranging the sides in this way does not change our perimeter because we've not made any sides longer or shorter, we've just put them in a different position.

It is, however, sometimes, significantly easier to just sum the side lengths.

Remember, summing the lengths of any polygon will give us the perimeter of that polygon.

That method always works.

It's just not necessarily the most efficient.

Well done today.

You've worked really hard.

I look forward to seeing you in our next lesson.