Lesson video

Hi, welcome to our lesson on the area of composite rectilinear shapes.

By the end of today's lesson, you'll be able to find the area of composite rectilinear shapes in a variety of ways.

Our lesson today has three parts, and we're going to start with the first one that deals with comparing areas.

It is possible to compare the areas of shapes without actually performing any calculations.

We can do this by considering the properties of our shapes.

For example, here I have two rectangles.

Which rectangle has the bigger area? How do you know? That's right, the rectangle on the right definitely has a bigger area.

We can see that both rectangles have the same height.

All that's different is how long the rectangle is.

Well, if one rectangle is longer than the other, and we know that to find the area of a rectangle, we multiply the long side by the short side, or the length by the width, base by the height, if both of them have one measurement the same, and for the other measurement, one is longer than the other, then the one with the longer measurement is going to have the larger area.

Notice that I didn't actually have any numbers.

I didn't actually carry out any multiplication.

What I did was I reasoned, and that in itself, was enough to tell me, which area was bigger.

It's time for your first task.

I'd like you to order the shapes from the one that has the smallest area to the one that has the largest area.

You'll notice I haven't given you any numbers.

I have marked particular sides of my shape though, and it's worth bearing in mind that any length that has just one line crossing it, are the same length.

So in other words, in A, where you can see a length that has just one line crossing it, all lengths in B and C that also have just one line crossing them, are the same length as that length in A.

Perhaps that will help with your reasoning.

Pause the video now while you complete the task.

Welcome back.

How did you get on with your reasoning? How did you order the shapes? Let's have a look.

C is the smallest shape, as it fits inside shape A, with some space left over.

We can split B into two triangles by cutting along the dotted line, and if we do that, and rearrange them, we can actually make the rectangle we can see in A.

and therefore we can work out that B and A have the same area.

So if you ordered your shapes by saying C has the smallest area, and A and B have the joint largest area, then you've got this right.

Do check your reasoning though.

Is it as strong as mine is? The area of a shape can be calculated by splitting the shape up and then calculating the area of each part, and that's what we did with our reasoning here.

When I cut that triangle in two and rearranged it, I made the same as the area of the rectangle.

Now let's just check that we understand that.

The area of shape A is the area of shape B.

Is it larger than, smaller than, or the same as? Pause the video while you work out which option it should be.

Welcome back.

Which one did you go for? It's actually C.

These two shapes, although they look different, actually have the same area, and you could have done this by cutting shape A, and rearranging two rectangles to make the rectangle on the right.

If you're not certain about that don't worry, because we're going to be looking at that particular technique in more detail in this lesson.

We're now going to look at three different ways to calculate the area of a composite rectilinear shape.

Remember, a composite rectilinear shape is a shape made of rectangles, which means if it's made of rectangles, we can break it back up into rectangles.

In what ways might we break up this shape in order to calculate the area? Well, we could cut straight across, as I've shown here with my dotted green line, that would leave me with three rectangles, and if I worked out the area of each rectangle, and then added these areas together, I'd have the area of my total shape.

I could also have cut here, and again I'd get three rectangles, different to my earlier ones, but again, by calculating their areas and then adding together, I'd find the total area of the shape.

I could also have cut here, and again, calculate the average rectangle, sum them, and I have the total area of the original shape.

Remember, a composite rectilinear shape is made of rectangles, so we can break it up into rectangles.

How might we break up this composite rectilinear shape? Well, we could cut it up vertically.

As you can see, I have two rectangles.

I could work out both of their areas and then add them together, but that's not the only place I could have broken up this shape.

I could have also cut here and got two rectangles.

Now, I could actually have done more than one cut and created more rectangles than just two, but that's more work and I want to try and be efficient if I can be.

So picking one of these two is definitely the way to go.

I've chosen to cut vertically.

Remember, it didn't matter which one I actually picked, because I'm going to get to the same answer either way.

So, let's calculate the area of each rectangle I can see.

Remember, I have to use the correct measurements.

Looking at the rectangle on the left, I can see that the two measurements that are the length and the width, or the base and height, are three and two.

So I want to multiply those two together to get the area of that rectangle.

The rectangle on the right, however, has a length and a width of four and three, so they're the two I want to multiply together.

I checked very carefully to make sure that I had covered the entire length and the entire width of my rectangle, that I wasn't using a length that was too short, or one that was too long because there's information in that composite rectilinear shape that I don't need to use here, and that's because of the cut I chose to make.

So let's work out the area.

Two lots of three, plus four lots of three, well, that's 6 plus 12 giving us 18.

It's now your turn.

You'll notice that I've given you the same composite rectilinear shape.

Look carefully at where that line is and what measurements are the length and the width for the two rectangles you can now see.

Pause the video while you work out the area of the composite rectilinear shape and check that it's the same as the area for where I made my cut.

Pause the video now.

Welcome back.

Did you get to the area being 18? Let's check the calculation in case you're not sure.

For the top rectangle, you should have been multiplying two and seven together and for the bottom rectangle four and one.

This means that our area is equal to 14, and 4 sum together, giving us 18.

Now that's just one method, but there are other ways to calculate the area of a composite rectilinear shape.

This method's referred to as completing the rectangle.

What do I mean by that? I mean this, I've drawn a dotted line across the gap in my composite rectilinear shape.

If I consider that that's a whole rectangle, what are the length and the width of that rectangle going to be? That's right, it's seven and nine.

If I multiply those two together, I have the area of that complete rectangle.

That area will be too big because I've added in this area that's not actually there.

Well, if I work out the area of the extra piece I added in, in this case the three by three square, I can take that away from the total area I just calculated and what remains is the area of my composite rectilinear shape.

So if I do seven multiplied by nine, the area of the completed rectangle, and then take away the area that I added in, so that's found by doing three times three, I can find the area of the composite rectilinear shape.

In other words, 7 times 9, or 63 subtract 3 times 3 or 9.

63 subtract 9 is 54.

And that is the area of my composite rectilinear shape.

This method can be really helpful for certain types of composite rectilinear shapes and part of your work later in this lesson is gonna be working out when it's useful.

Now, we can also rearrange a composite rectilinear shape into a rectangle.

Now we've done cutting up and rearranging shapes before.

For example, in the first section today, we cut that triangle in half and rearranged it to show that it had the same area as the rectangle.

So can we do something similar here? And we can.

If I cut here and take that top rectangle, and move it into the gap that's below, I've made a complete whole rectangle.

So I can just calculate the area of that shape.

In other words, 5 multiplied by 7, the area's 35.

I've done that by rearranging and it was so fast.

Again, sometimes this method might be preferential to use, but is it always? When is it good and when is it not? We'll be thinking about that very soon.

In fact, we're thinking about it here in our second task.

I'd like you to calculate the area of this composite rectilinear shape using each of the three methods we've looked at so far.

Remember, that's method one, breaking the composite rectilinear shape up into rectangles, method two where we draw to complete the rectangle, work out the total area and then subtract the extra area that we added in, and method three where we can cut the composite rectilinear shape and rearrange by moving the rectangles around so that we can form a new rectangle, and calculate the area of that.

What I'd like to know is once you've done that, which of the three methods was the most efficient and why do you think that? Pause the video while you have a go at this task.

Welcome back.

Let's see how you got on.

This is my working for the first of the methods where I cut the shape up.

Now you could have cut in a different place and that's fine too.

I just happened to have chosen this way.

So for my particular cut, I have a rectangle that has a length and a width of two and nine and another rectangle that's nine and five.

So my area is found by doing two times nine add 5 times 9, giving us 18 plus 45 or 63 in total.

Now, you might have thought that this method was the most efficient as it's the one you find the easiest.

So you think it's easy, it's the one you'd always like to use.

That's good reasoning if that's the one you picked.

In this method, we asked you to complete the rectangle, so if I complete the rectangle, I can see that the whole rectangle has a length and a width of 14 and 9.

Now the extra rectangle that I've added has a lengthen a width of 9 and 7.

So to work out the area of the composite rectilinear shape, I need to do 14 multiplied by 9, subtract 7 multiplied by 9, or 126 takeaway 63, which leaves us with an area of 63.

You might have said this method is the most efficient because it involves less work when the composite rectilinear shape is more complicated, and that's perfectly fine if that's the one you've gone for.

Our final method is this one, where we make a cut and we rearrange the shape.

So you can see where I've placed my cut, the rectangle that's two units by nine units, I'm suggesting that you slide it round so it goes adjacent to the nine by five rectangle.

If I do that, then the length of two goes next to the length of five, making a total length of seven.

I would therefore have a length of seven units along one side of my rectangle and a perpendicular distance of nine.

Multiplying the two of these together therefore gives me 7 times 9 or an area of 63.

You might have said that this method is the most efficient as it made just one rectangle that you had to find the area of, so less working.

And again, that's great if that's the one you went for.

Let's move on now to our final section where we evaluate those three methods in more detail.

Here's the working from our three methods.

So method one, where we broke the composite rectilinear shape up into rectangles is the first line.

The second line is our method where we completed the rectangle and the third line of working, comes from when we rearranged the composite rectilinear shape.

As you can see, we've got the same area each time, but let's look at the working.

Does that working look familiar? Is there something similar between each set? That's right, in the first line, I have two lots of nine added to five lots of nine Well in total, therefore, that's seven lots of nine.

In the second line, I had 14 lots of 9 and I took away 7 lots of 9, which left me with 7 lots of 9.

And in fact the very bottom line was two and five added together and then multiplied by nine, so seven lots of nine.

In fact, it's showing that all three lines of working are in fact equivalent calculations, which is why any of our methods will work if we apply them correctly.

Now, a quick check.

Our different methods give us different answers.

Is that true or false? And then make sure you justify your answer.

Pause the video now while you make your selection.

That's right, it's false.

Our different methods didn't give us different answers at all.

You can use any method, but some might be easier depending on the shape.

In our final task, I'd like you to draw your own composite rectilinear shape and then try to find the area using the three different methods.

Can you draw a shape where one or more of these methods is not helpful? So in fact one of the methods might be superior to the others for that particular composite rectilinear shape.

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

Welcome back.

Now, the shapes you've drawn will vary, but they should all be built from rectangles placed adjacent to each other.

Here's an example of a shape where completing the rectangle is not going to be particularly helpful.

If I complete this rectangle, the area that I've added in is another composite rectilinear shape.

That doesn't seem like that's helping me particularly.

And trying to rearrange this doesn't easily lead to another rectangle.

I can't see a part that fits nicely onto what's left.

This is definitely an example of a shape where breaking it into rectangles, working out the area of each and then summing them together is almost certainly going to be easier and faster.

Let's summarise what we've done today.

A composite rectilinear shape is made of rectangles and the area of a composite rectilinear shape can be found in a variety of different ways.

We saw three of those ways today.

Depending on the shape, a particular method may be more useful than others or it might be that any of our methods is equally useful.

It is important to know that there are multiple approaches, however, because there may be shapes where it's significantly easier to use one method over the other.

Well done.

You've worked really hard today and I look forward to seeing you in our next lesson.