Lesson video

Hi, welcome to today's lesson on calculating missing side lengths from the area of a composite rectilinear shape.

By the end of today's lesson, that'll be exactly what you're able to do.

We're going to start with section one, which is calculating area with missing side lengths.

In other words, you're going to find the area of different composite rectilinear shapes, but not all of the side legs are initially going to be on the diagram, so you're going to need to work those out first.

A composite rectilinear shape may not have the length of every side shown, so as just mentioned, in those situations, you may need to deduce the missing side length in order to find the area.

Can we calculate that missing side length? How do we do it? That's right.

We can see that the two side lengths to the right of the shape, so that's the seven and the side we don't know, they are both parallel to the vertical side labelled 15.

Now, they're parallel to the line, and they don't overlap with each other, which means if we placed the seven centimetre line on top of the line that we don't know, look what happens.

That's right.

They're exactly the same distance as the 15 centimetre line.

And we saw reasoning like this in our previous lessons.

Do you remember the red line that was inside the rectangle, and we asked you how long that red line was, and we talked about those sides moving to the right and moving down.

Same argument here.

We can think about that side moving and compare it to another length.

In this way, we've been able to work out the distance that's been missing.

So what is that missing distance? Well, 7 add something is 15, or we can think about that as 15 takeaway 7 leaves us what? That's right, it leaves us with eight.

So our missing length is eight centimetres.

Here I have a composite rectilinear shape, and you can see that there's quite a lot of measurements on it.

I could work out that missing distance there because I know that the entire height of the shape is six, and I could see that from the bottom up to the gap was 2.

7.

So what did I need to get to six? I needed another 3.

3.

Now, why have I worked that out? Can you see on my diagram that I've added a dotted line across the top of my shape? We looked at the different methods for calculating the area of a composite rectilinear shape in the previous lesson.

Which of the three methods do you think I'm going to use here if I've added that dotted line and added in the length of 3.

3 centimetres? That's right.

I'm going to be using the second of the methods we looked at in that lesson, where I complete the rectangle, find the area of the whole rectangle, and then take away the part that I added in.

Let's see that working now.

To find the area of the composite rectilinear shape, I found the area of the whole rectangle if it was complete.

So I did six multiplied by eight.

And then I took away the extra area that had been added in, so two lots of 3.

3.

I can now work out that calculation.

6 times 8 is 48, and two lots of 3.

3 is 6.

6, so 48 subtract 6.

6, leading me with 41.

4.

Can you see the composite rectilinear shape on the right? It's your turn now, and you need to calculate the area of this shape.

It's up to you which of the three methods we looked at last lesson that you would like to use now.

You could do what I did and complete the rectangle if you wish, but you will need to work out a missing side length.

You could break the composite rectilinear shape into rectangles, and it's up to you, remember, where you choose to break it, but if you do, you will still need to work out some missing lengths.

You could try for rearranging if you wish, if you think that one's going to be the easiest.

Pause the video now while you find the area of this composite rectilinear shape.

Welcome back.

Which method did you go for? Well, I'm going to talk you through one particular method.

I'm gonna be looking at splitting it into rectangles.

Now, the reason I've chosen this one is because I can see two obvious ways to split this shape, and I want to talk about which one would be better.

Now, you could have split the shape in the way you see on the screen right now, but if you do that, you can see the two extra arrows I've added in, in order to work out the area of the smaller rectangle, I need to first work out both of those missing side lengths.

So that's two extra bits of information I have to find.

What would happen though if I split the composite rectilinear shape in a different way? By splitting it up this way, there is only one missing side length I need to calculate.

I don't know about you, but that to me seems more efficient, only having to find out one thing rather than two, sounds better to me.

It's absolutely fine if you found the area by breaking the composite rectilinear shape up in a different way.

But one of the things we're always keen on as mathematicians is to look at how we could have been more efficient.

So it's worth evaluating both of these approaches to see which one produces less work.

So I'm now ready to find the area of my composite rectilinear shape.

Remember we need to work out the area of both rectangles, and then sum these areas.

The area of the larger rectangle is found by multiplying 2.

6 by 5.

Remember, we don't want to use the six length.

The six length goes past our rectangle.

We need to use the correct measurements for each rectangle.

So 2.

6 multiplied by 5, and then the top rectangle is found by multiplying 1 and 4 together.

2.

6 times 5 is 13, and 1 times 4 is 4.

So 13 add 4 gives us an area of 17.

It's now time for your first task.

I'd like you to calculate the area of each composite rectilinear shape.

It's entirely up to you which of the three methods you use.

If you like, feel free to use a couple of the methods, so that you can evaluate which one is superior for each question.

You may find that one question benefits one particular method, and the other benefits from a different method, and that is something we talked about in our previous lesson, is working out which method is best when.

What properties of a composite rectilinear shape favour one method over the other.

Think about this as you have a go at task one.

Pause the video now while you have a go.

Welcome back.

How did you get on? Let's go through the working so you can see how your work compares to mine.

In a, I opted for calculating a missing side length and doing the complete the rectangle option.

You could of course have gone for breaking this into two rectangles, finding the area and summing them.

I went for this particular method, because then I only had one length I needed to calculate so I felt this was easier.

Again, though, it's your choice.

The entire rectangle would have an area of 16 multiplied by 12, but then I've added in the extra rectangle, which is 11 by 4.

So to work out the area of the composite rectilinear shape, I have done 16 multiplied by 12, and then subtracted 11 multiplied by 4, leaving me with a total area of 148.

Let's look at b now.

In b, I absolutely went for method two, where I found the area of the entire composite rectilinear shape and then subtracted that extra area that I'd added in.

We can see that my shape has a gap in the middle.

I definitely don't want that, and I didn't wanna break this into rectangles going around that gap, so that's why I've opted for method two.

In order to find the area of the entire shape, I've done 17 multiplied by 19, but I then needed to take away the extra I'D added in.

I can see in that rectangle inside my shape that above it there is a gap of three and below a gap of five.

That makes a total of eight.

The entire height of that shape is 17, so what's left over? Well, 17 takeaway 8 is 9, and that's where the 9 for the inner rectangle has come from.

Similar reasoning lets us work across the rectangle.

I can see on the left, a gap of two, and on the right, a gap of 11, making a total of 13, but the entire shape has a length of 19.

Well, 19 takeaway 13 is 6, which is why I have a 6 horizontally in that inner rectangle.

So that means that the area of the inner rectangle is found by multiplying 9 and 6 together.

Now that I've got those two multiplications, I need to subtract.

So the total area is found by doing 17 multiplied by 19, take away 9 multiplied by 6, or in other words, 269.

It's now time for the second section of our lesson, and this is where we'll be finding a missing side length when we are told what the area of the composite rectilinear shape is.

So it's similar to what we were just doing in that we still need to calculate a missing side length, but this time we're given the area and we're working backwards to see what the missing side length has to be.

Now, it does depend on what side length we're being asked to find.

We do need enough current information to know for definite.

So given the area, it could be possible to calculate a missing side length assuming we have enough information.

Let's consider this composite rectilinear shape.

I've said that the area is 54, and I've put some measurements on, and what I want to do is find that missing measurement across the bottom.

Which of my three methods for finding the area of a composite rectilinear shape do you think is going to be most useful here, and why? Pause the video and take a moment to reflect before we continue.

Welcome back.

Which of the methods would you go for? Personally, I'm choosing method two, and the reason for that is I know that if I complete that rectangle, then to find the area I would do 5 multiplied by X, take away the area of the extra rectangle I've added in.

Now I can see that extra rectangle has a length of two, but I don't know the other length at the moment, but I can calculate it.

If the entire height of that shape is five, and I can already see I have a length of two, what's the remaining length going to be? Well, it has to be three.

In other words, the first thing I did when starting with this problem was to think what measurements can I already deduce? So before I start calculating, what information did I already know or could I already work out and add to my diagram? Now I can start to see method two forming on the screen more clearly.

Let's go to our working.

The area of the composite rectilinear shape is found by finding the area of the whole rectangle.

So five multiplied by x.

Take away the additional area I added in.

So two multiplied by three.

I can tidy that up a little bit so that 54 is equal to 5 lots of x takeaway 6.

Something takeaway 6 is 54.

Something must be 60, because 60 takeaway 6 is 54, so 5x must be equal to 60.

What do you multiply 5 by to get to 60? Well, we know our 5 times table, so it's 12.

In other words, our missing length is 12 centimetres long.

Now, this is the great bit about maths.

When we get to an answer, it's often possible to go back and to check that, and this is exactly what I can do now.

I can check that method by saying 5 multiplied by 12, take away 2 multiplied by 3.

5 multiplied by 12 is 60, 60 takeaway 2 multiplied by 3, so 6, so 60 takeaway 6 is 54.

Perfect, I've got back to that area.

This tells me I haven't made a mistake.

And that's a great skill to have for maths.

If you can check your work and you understand what you've done, you can make sure that you find any areas nice and quickly.

It's now your turn.

Here's a composite rectilinear shape, and you're told the area is 18.

4.

I've shown you what missing length I'd like you to calculate.

Remember what I did first.

Work out any measurements that you do know already.

Then think about which of the methods for finding the error of a composite rectilinear shape is going to be the most useful to you here.

Pause the video while you have a go at this question.

Welcome back.

Let's see how you got on.

Did you deduce that that missing side length had a length of one? You could work that out because you knew the total was four and the bottom had three.

So what was the missing amount? You can't work out anything vertically at the moment, because although we know the total is six, we have two missing side lengths, one of which we're trying to calculate, but at least we've been able to put something on the diagram.

Can we see now how we might break this shape up in order to calculate the area, and therefore which method is going to give us the most useful working care? I've broken my shape into rectangles, because now I can see the total area of the composite rectilinear shape can be found by multiplying six and three together, that's the area of the large rectangle, and adding on the area of the small rectangle, which is found by doing one multiplied by y.

That seems quite nice.

So 18.

4 is equal to 1 multiplied by y, or just y, and adding on 6 multiplied by 3, or 18.

In other words, 18.

4 equals 18 plus something.

Oh, well, that'll be 0.

4 then.

So that missing side length is 0.

4 centimetres.

That got a lot easier because we knew that the length of one was there.

So by deducing what we already knew and adding it to the diagram, we made this a lot easier than it would've been if we hadn't worked that out first.

Remember, that's our key steps.

Deduce what you can, add it to the diagram, then evaluate and work out what you think the best approach is going to be.

It's now time for our second task.

And you'll see that these shapes may look familiar.

That's because these shapes appeared in task one, only I've given you all the measurements you needed, and you only had to work out a couple, maybe even just one.

This time I've hidden one of the side lengths, and I said I want you to calculate that missing side length.

So think carefully about the method you used to work these out the first time.

You're gonna be able to use that method again, but remember, the numbers are different, and you're working to find a missing side length this time.

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

Welcome back.

Let's have a look at a.

So the first thing I did was I deduced the missing side lengths that I could.

I could see that I have a total of 18, and one of the parallel sides equal to 7.

So I can work out the other parallel side by saying if the total is 18 and I've already got 7, what remains, it must be 11.

Having done that, I then broke my composite rectilinear shape into two rectangles.

I did 23 multiplied by 11, and I added that to 7 multiplied by h.

I then said, well, 23 multiplied by 11 is 253, and 7 times h is just 7h.

What do you add to 253 to get to 309? You add 56.

So 56 must be the area of the rectangle 7 multiplied by h.

Therefore that missing length must be 8.

In b, remember, we'd opted for the method of calculating the area of the entire shape and subtracting what was inside.

So if the area is 187, I know that the area of the whole rectangle would be found by multiplying 13 by t, and then I'd subtract that inner rectangle.

So four multiplied by two for that area.

In other words, 187 is equal to 13 lots of t, takeaway 8.

That means 13t must be equal to 195.

195 divided by 13 is 15.

So that marked side is 15 centimetres long.

It's now time for the final section, which is working with perimeter in area.

In other words, using one perhaps to calculate the other.

Let's see.

Different composite rectilinear shapes may have the same perimeter area yet look completely different.

What do you notice about these two shapes? Is their area the same? Is their perimeter the same? What do you think? Correct, their area is definitely different.

We can see that the shape on the right is taking up less space than the shape on the left.

What about their perimeters however? That's right, their perimeters are the same, and we can see that by either moving a couple of the sides to show that you can get to the rectangle on the left, or we can see that the top side and the two bottom parallel sides are the same length, just like they are in a rectangle, and the same is true with the vertical sides.

So, same perimeter, but different areas.

It's now time for our final task.

You can see four shapes here.

I'd like to know which of a, b, c, and d have the same area, and which shapes have the same perimeter.

You may be able to work out quite a lot just by deduction.

So without having to do any actual calculations, you may already have some answers.

Don't hesitate though.

If you want to do calculations, you are very welcome to do so, and you may find that you need to, but try to see how much you can reason.

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

Welcome back.

Did you spot that shape a has an area of 80? b, c, and d all have areas smaller than that, because part of the whole rectangle is missing in each case.

But what's interesting is the area that's missing in each of those is the same area.

In b, you can see there's a missing square of three by three.

The same is true in c, and the same is true in d, which means they're all the same rectangle with the same amount taken away each time, which means they must all have the same area.

And you can work this out by doing 8 multiplied by 10, subtract 3 multiplied by 3, giving it an area of 71.

What about perimeter? Well, a and b definitely have the same perimeter.

We can see that by just rearranging the sides.

In fact, they both have perimeter of 36.

C, on the other hand, we already knew was going to have a larger perimeter, because we can see that two of the sides that we would need to move overlap each other.

So they can't just be rearranged into a straight rectangle.

There's a bit more to go.

And in fact, the extra amount is two lots of three, because that's the two vertical sides that come down into the shape.

Now, d is an interesting one.

We say that d is ambiguous because we're talking about the perimeter of a shape.

Would the perimeter include that inner square or not? What do you think? When calculating perimeter or area for composite rectilinear shapes, it can be more efficient to consider manipulating the shape, which is what we were just talking about.

Now, let's just match our words to their definitions to check our understanding of these three words.

We've been using them a lot, so it's important to go back and check that we are fully happy with their full definitions.

Please match the words on the left to the definition on the right.

Pause the video now while you do this.

Welcome back.

Let's see how you match them.

Area is the size of the surface.

Perimeter is the distance around a 2D shape.

So, surely that means that length must be the measurement between two points.

And it is indeed, except wait a minute.

You can also describe the distance around a 2D shape as a length.

If a length is a measurement between two points, then the distance around a shape is surely the measurement between where I started and when I got back to where I finished.

In other words, going around my rectangle.

And in fact, perimeter is the special name given to the length around a 2D shape.

Oh, how exciting.

So perimeter is just a very special kind of length.

I thought you might enjoy that.

It's a slight subtlety that isn't always covered, and I thought how exciting to be able to address that now.

I thought you might like that one.

Did you spot that length could go to two? Let's summarise what we've covered today in our lesson.

A composite rectilinear shape may not have the length of every side shown.

In these situations, you may need to deduce the missing side length in order to calculate the area.

And that's what we did in the first section of our lesson today.

Given the area of a composite rectilinear shape, it may be possible to calculate a missing side length, and that's what we did in our second section.

In section three, remember, we brought this together to do some reasoning about perimeter and area with composite rectilinear shapes.

Well done.

You've worked really well today.

I look forward to seeing you in our next lesson.