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I can calculate the area of compound shapes made from rectangles, triangles, parallelograms and trapeziums efficiently.

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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

There are multiple ways to find the area of compound shapes.
The most efficient method may depend on the features of the shape.
A method is to break the shape into shapes you can find the area of.
A method is to complete the shape.
A method is to rearrange the shape.

Common misconception

Pupils may multiply all given lengths.

Remind pupils that area is found by multiplying two perpendicular lengths from the shape and in some cases a coefficient (half for a triangle, pi for a circle). Trapeziums use the mean length of the parallel sides (one length).

Keywords

Area - The area is the size of the surface and states the number of unit squares needed to completely cover that surface.
Compound shape - A compound shape is a shape created using two or more basic shapes.
Composite shape - A composite shape is an alternative for a compound shape.

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Hello, I'm Mrs. Lashley and I'm gonna be working with you as we go through the lesson today.

I hope you're ready to make a start.

So our learning outcome today is to be able to find the area of compound shapes efficiently and the compound shapes will be made from rectangles, triangles, parallelograms, and trapeziums. So let's make a start, on the screen there are some words that you will be familiar with, but you may want to pause the video so that you can reread them and make sure that you're happy before we move on to the main part of the lesson.

So our lesson is broken into three learning cycles and we're gonna be looking at three different methods of how to find the area of compound shapes in each of those cycles.

So our first learning cycle is looking at the method of subdividing the compound shape.

So let's make a start with that one.

So to find the area of a compound shape, we can split it into standard shapes that created it.

And that's the definition of a compound or a composite shape is a shape that is created by two or more basic shapes.

So here we have a rectilinear compound shape and it could be split horizontally into two basic rectangles, but it also could have been split vertically into two rectangles.

There's another way that we could split that up.

Can you think of it? So it could be split both vertically and horizontally, and then we have three rectangles.

The reason this method works and by splitting it into basic shapes is that the rectangle has a formula for the area.

So we know that the area of a rectangle is length times width or base times perpendicular height.

And that way we'll be able to build our area of our compound shape.

So using the horizontal split, we can calculate the area of each of the rectangles and then add them together or sum them.

So the area of rectangle A, the one that's marked with the A is eight multiplied by three, which is 24 square metres.

And hopefully you can see where the eight and the three have come from from the diagram.

So now we need to work out the area of rectangle B and that is seven multiplied by four and that gives us 28 square metres.

So the two basic rectangles that have created this compound shape have got an area of 24 and an area of 28 respectively, which means we can sum them together to find that the area of the compound shape is 52 square metres.

So here is a check, another rectilinear compound shape.

There are some blanks in the working.

There are some spaces for you to work out which numbers go there using the diagram to support you, so pause the video whilst you're working that out and then when you're ready for the answers, press play.

So rectangle A was six metres wide and it would be 14 metres long.

So that did need you to recognise that there was a vertical length of nine metres, but that isn't the full edge of that rectangle.

So we need to go up an additional five metres, which gives us the 14, six multiplied by 14 is 84.

Area of rectangle B is seven multiplied by five.

And you can see the dimensions there quite clearly in the diagram.

So what is the product of seven and five? Well, that's 35.

And so the area of the compound shape is the sum of those two numbers, 84 plus 35.

If you did 35 plus 84, we still get 119 square metres.

So let's look at a more complicated compound shape.

So previously they were rectilinear compound shapes, which means that they were created with rectangles only.

But we know that a compound or a composite shape is made from basic shapes and those basic shapes could be a rectangle but might be triangles, parallelograms or trapezius.

So this compound shape is symmetrical both horizontally and vertically.

So it has got lines of symmetry.

How can it be split into standard shapes to help find the area.

So just think about that.

Where would you split this compound shape up in order to be able to calculate the area in a similar way to how we just did the rectal linear compound shape? Well, perhaps you just split it here and from here we get two trapeziums and one rectangle.

We don't know if it's square or not.

So it's a rectangle, but perhaps you split it like this instead.

So we have four triangles and a rectangle in the middle.

What do we know about those triangles? Well, we know those triangles will be congruent and the reason we know they're congruent is because of the symmetry of this shape.

So knowing that this shape is symmetrical allows us to know that though all four of those triangles are actually congruent to each other.

So whichever way you decided to split, there is some further information that we need before we can actually calculate the area.

Can you see where that is? So here are the two ways that you may have decided to split this shape.

You could have split sort of a combination of the two and then have more areas to work out.

But we do want to try and be efficient.

So we want to sort of limit as many areas as possible when we're splitting it up so that we're not having to do too many calculations.

So what information is still required? Well, on the left hand one, the one where we've split it into two trapezium and a rectangle, we need the length of the parallel sides of a trapezium and that's marked with A there.

So that is the length, which is the edge of a trapezium, but it's also the edge of the rectangle.

It's a really important length for us to know in order to be able to go on and find the area.

On the second way of dividing the shape, we need to know the length of the base of those triangles.

We know that will be the same in all four because of the symmetry, but we don't have its base and the area of a triangle is half times base times perpendicular height and therefore we need to work out the base.

Once we have that base, we could use that then to find the width of the rectangle as well.

So knowing the base will allow us to get the rectangle's width.

So both lengths A and B can be found in the same way.

So I'm using this division of the shape and both of these lengths can be found in the same way.

So we can see that A plus 2B equals 30.

So we were given that this sort of maximum vertical height of this shape was 30 centimetres.

And because of the symmetry we know that both of the triangles will have this length B at the base.

So that's the two B part and the A is the width of the rectangle.

We also know that the perpendicular height of the triangle or the trapezium, if you're doing the other way, the split, let's label that H, means that two of those, again, the symmetry of the shape means that the other on the right hand side, I could have labelled the same distance as H.

And there is a 15 centimetres in the middle.

But we have the maximum width of the shape is 63 centimetres.

So we can make ourselves a linear equation to say that two lots of our H, which is either the perpendicular height of the triangle or the trapezium, depending on which split, plus the 15 has to sum 63.

So we can solve that one because there is only one variable.

So we can solve that to find the H is actually 24 centimetres.

So we now have the perpendicular height of the triangle or the perpendicular height of the trapezium of which we need in both of their area formulae, we need it for the triangle and we also need it in a trapezium.

As the triangles are right angled, then we can set up Pythagoras theorem.

So the 26 centimetres has been labelled on the external for the bottom right slant of the triangle, but they're all marked with a hash.

So we know that all of those are the same.

We know that from the symmetry as well.

26 centimetres is the hypotonus, 24 centimetres is one of the edges of the triangle, and B is our base edge that we're trying to calculate.

So we can evaluate our squares.

26 squared is 676.

You're not expected to know that, you can use a calculator or calculate it by hand.

Doing 26 multiplied by 26, 24 squared is 576.

So by rearranging B squared is 100 and you probably should know this one, that that would mean that B is 10.

So now that we have the base of the triangle, we can use our original equation of A plus 2B equals 30 to work out the width of the rectangle.

So given that B is 10, A plus two lots of 10 equals 30, therefore A equals 10.

So this diagram is not drawn accurately because they look different in terms of the dimensions, but we've calculated they are actually equal.

So now that we have all the required information, we can calculate the area.

So on these more complicated compound shapes, it might be that you need to do a little bit of maths to get all of the required information before you can get the area.

So we've got everything we need using this split.

The shape is divided into four congruent triangles and we've spoken about why that is.

And one rectangle.

Here's a little sketch of the triangles and the rectangle with the dimensions that are needed.

So the area of one triangle is half times base times perpendicular height, base being 10, we calculated that and 24, we worked that out from the given information.

So that equals 120 square centimetres, which means that all four, there are four congruent triangles, would have an area of 480 square centimetres and the rectangle would be base times perpendicular height or length times width.

And so that's 630.

So we can say that the compound shape, the total area of the compound shape is 1,110 square centimetres.

So for your check here is a compound shape, I'd like you to match the areas to the correct shape.

So I've split it, I've divided it into three basic shapes.

I've given you the three different areas.

I just want you to match up which area goes with which part of the compound shape, press pause whilst you're doing that and then press play to move on.

30 square centimetres is the area of rectangle B.

We know that it's five centimetres and we can calculate that its height would be six by doing 10, subtract four, the trapezium would be 24 square centimetres.

You need to have the parallel sides.

Sum them, multiply it by the distance between and half it to work that one out, but it might have been process of elimination that got you that one 'cause C is another rectangle, so seven times 10 is 70, so we know that C matched with the 70.

So the next part of this check is hence, what is the area of the compound shape? Pause the video whilst you work that out and then press play to check your answer.

So that's just summing the three areas.

So 30 plus 24 plus 70 is 124 square centimetres.

So we're now up to task A where you are gonna do a bit of practise of subdividing compound shapes to find the area.

So there are two compound shapes.

These are both rectilinear compound shapes and you need to find their areas.

So pause the video whilst you do that and then when you're ready for question two, press play.

So here is question two.

Again, you're subdividing the compound shapes, so splitting them into basic shapes where we do have a formula for finding the area.

It may be that you need to work out some further information, some more required information necessary before you can get the area.

So pause the video and work through parts A and part B.

And when you press play, we're gonna go through the answers to this task.

Okay, so here's part A, this is an example.

I've split it vertically.

You may have chosen to split it horizontally or split it in more ways, but your area should be the same as this.

So the area is 55 square metres.

It may be that you want to pause the video to check my working out if you did it a different way and got it incorrect, but hopefully you got it correct and well done.

Part B, and again this is an example of one method.

So I've split this vertically, I've got three rectangles and the total area is 22.

5 square centimetres.

Question two, I've divided it into a trapezium on the top and a rectangle.

You may have done more divisions, you may have got rectangles or triangles and I am working this out.

So the area of A is half times the sum of the parallel sides times the perpendicular height, which gives you 18.

Area of B is a rectangle, using the hash marks, we know the dimensions there.

And so it's 16.

2 and the total area is 34.

So once again, you may have done this in a slightly different way.

It depends on how you decided to divide the shape, but our area should be the same.

Part B, an example of a method.

So A is 32 square centimetres, B is a trapezium and that's 38 square centimetres.

The overall total area is 70.

You may have done that a different way to the way I've done it, but we should have both got the area of 70.

So now we're onto the second learning cycle.

Remember we are looking at methods to find the area of compound shape and this method I'm calling completing the shape and we're gonna work through this now.

So to find the area of a compound shape, we could subdivide it, we saw that previously, but we could complete the compound shape to form a standard shape and then subtract any additional area.

So let's look at this one on the screen.

It's a rectilinear compound shape and what I can do is I can complete the shape.

And what I mean by that is I've given it additional area by now forming a standard shape.

The shape that I have formed is a rectangle, but clearly the area is larger than the area of the compound shape and that's because I've given more area to it.

That rectangle that I've added on needs to be subtracted off.

So a completed shape would be a rectangle with dimensions of 10 metres by eight metres.

And so that would have an area of 80 square metres.

We know we just multiply those to get the area of a rectangle.

The addition of the rectangle is four metres by seven metres.

So that was an additional area of 28 square metres.

So I need to remove that additional 28 because I added to the area.

So the compound shape, the question I was trying to answer has an area of 52 square metres because I've got the 80, which is the completed rectangle and I'm subtracting the additional area to make the rectangle.

So here's a check.

Which of these have successfully completed the shape? So pause the video and then when you're ready to check, press play.

That would be C.

So A and B, there has been additional area added to it, but it hasn't made a standard shape.

And the whole point of this method is to make a standard shape because we have a formula to work out the area of that standard shape and then we can start subtracting off what we've added to get to our compound shape.

So we haven't made a standard shape on A or B.

Here's another check for you.

The area of the rectangle which surrounds the shape is 99 square centimetres.

What is the area of the cross shape, which we could call a Dodecagon because it's got 12 edges.

So pause the video whilst you're calculating only the area of the cross shape using the completed shape to support you.

Press play when you're ready to check your answer.

So the additional area, we completed the shape by making it into a rectangle by adding four squares at the corners.

And all of these would be congruent to each other.

We know that because of the hash marks and they would be three centimetres by three centimetres.

So they have an area of nine.

So we need to take the rectangle, 99, and subtract four lots of nine away, which leaves us 63 square centimetres.

Well done if you manage to get that.

So let's look at the same shape as previous, but trying this method.

So more complex shapes can also be surrounded to complete a rectangle.

So we've got this horizontal and vertically symmetrical shape and I've created a rectangle by adding in some additional area.

So two congruent isosceles trapeziums have been added to complete the rectangle.

So we will need to remove their area to get the area of the compound shape.

The perpendicular of these isosceles trapeziums can be calculated using Pythagoras theorem.

And when you do that it is 10 centimetres.

So the area of a trapezium can be calculated by using the formula A plus B, where A and B are the parallel edges multiplied by H, where H is the perpendicular height and then divided by two.

So here is the isosceles trapezium that we are trying to get the area for.

So we need to identify our parallel edges, that's the 15 and the 63.

Our perpendicular height is the 10 which we would have calculated using Pythagoras theorem.

So the area for this is 390 square centimetres.

So going back to our shape, we need to work out the area of the rectangle, that's the completed shape, which is 1,890.

And then we need the air of the two trapezium.

We just calculated that one of them is 390, so multiplying it by two 'cause there are two of them gives us 780.

So the compound shape will be the area of the rectangle.

That was us completing it and then subtracting the area of the trapezium.

So our area is 1,110 square centimetres.

A check for you, a triangle has been added to make the compound shape a rectangle.

Hopefully you can see that triangle in the top left corner.

I'd like you to complete the perpendicular lengths of the triangle.

So there are two perpendicular lengths, length one and length two.

So pause the video whilst you're working those out, press play when you're ready to check.

So using the given dimensions on the compound shape, we can see that if it's a rectangle, we know the opposite edges are equal.

So 25 subtract 21 would give you four and 32 subtract 26 would give you the six.

It doesn't matter which one you called length one and which one you called length two, as long as you got four and six as your two lengths, they are perpendicular.

We know that they are perpendicular because it's a rectangle.

There would be a right angle in that top left corner.

So task B, by completing the shape.

So remember that's making it into a standard shape, normally a rectangle, and then subtracting additional areas back off of it to get the area of the composite shape.

So part A and part B, pause the video whilst you're working through question one.

And then when you press play, we'll move on to question two.

So question two, there's two parts to it.

Part A, by completing the shape, find the area of this composite shape.

And then part B, if you had subdivided the shape to find the area, would it have been more efficient? So I want you to reflect on the two methods that we've seen so far this lesson.

So part A, complete it.

So surround it by a standard shape and then subtract off the additional areas.

And part B, consider if you had done it by subdivision, would it have been more efficient than completing? Press pause whilst you're working through question two.

And then when you press play, we'll move on to question three.

Question three, Izzy says, \"I think it is always more efficient when finding the area of a compound shape to divide the shape up, to subdivide it, rather than use the method of completing the shape and subtracting the additional area.\" So you might be in agreement with Izzy or maybe you disagree with Izzy, but I would like you for question three.

Part A is to sketch an example of a compound shape to agree with Izzy.

A shape where subdividing it is more efficient than completing the shape.

And then part B, I want you to think of examples of compound shapes to disagree with Izzy, where completing the shape is more efficient than dividing it up.

And it may be that some of the practise tasks you've already done can help you with coming up with some of those answers.

So pause the video whilst you think about that.

You may wanna discuss with somebody close by on this one.

See if you agree with which shapes are more efficient one way than the other.

And then when you press play, we're gonna go through our answers.

So question one A, you needed to complete the shape to find the area of the composite shape.

So by surrounding it to make a rectangle, the area is 576, it's a 32 centimetre by 18 centimetre rectangle.

And the additional area is a triangle.

We have all of the information necessary.

So we know that the base is 18 and its perpendicular height is 14, so half times base times perpendicular height gives you an area of 126.

Therefore the compound shape has an area of 450 square centimetres.

Moving on to part B, this one we have two additional areas that we would need to subtract.

When we surround it, we make a triangle at the top and an extra rectangle.

So the surrounding rectangle has an area of 120 square centimetres because it is a 12 by 10 rectangle, the area of the triangle is 30, we can calculate that the perpendicular lengths are 12 and five.

So then half times base times perpendicular height gives you the 30.

And the rectangle that sort of fills in the gap is a five by two, which is 10 square centimetres.

So now we can find the area of the compound shape is 80 square centimetres.

Question two, you part A, you needed to complete the shape.

So by completing the shape we've added a triangle and two rectangles.

So the surrounding rectangle has an area of 125 square metres.

The area of the triangle has an area of three rectangle one, which is the one on the left is 34 rectangle two, which the one on the right is 17.

So the area of the compound shape is 71 square metres and that's by taking the surrounding rectangle and subtracting the three additional areas away from it.

Part B was then to reflect on that and say if you had subdivided, would it have been more efficient? So it can be subdivided into two shapes, which is a trapezium at the top and a rectangle.

So only two area calculations would be needed compared to the four when we completed the shape.

So subdividing would be more efficient.

And now on to question three where you had to come up with examples to agree and to disagree.

So these three that I have come up with are more efficient to subdivide.

So if we look at the one that looks like an arrow, I would split that into a rectangle and a triangle.

Whereas if I surrounded it, I'd have a rectangle at the top of rectangle at the bottom and two triangles.

So that would be five areas that I would need to calculate compared to two.

And that's similar on the other ones.

You may have come up with different ones for that.

In my opinion, they are more efficient if you use the subdivision method than if you complete the area.

Part B though, if I was trying to disagree with Izzy, then I've got the one where it's just one triangle missing at the top.

If I split that, I'd need to split that maybe into a trapezium and a rectangle.

Whereas a rectangle and a triangle is the same amount but I just feel it's more efficient.

Completing the shape with an L shape.

Again, two areas to find and completing the one that looks a little bit like a flag, again in my opinion, are much more efficient to complete it and subtract the additional area.

And that's because the additional area's just one area, I think once there becomes more areas then it becomes a little bit less efficient.

We're now up to the last learning cycle, which is another method to find the area.

And this one is rearranging the area of the compound shape.

So sometimes it might be possible to rearrange the area and simplify the calculation.

So here we've got our recline compound shape.

This section can be moved to here and now our rectilinear compound shape, rather than the being two separate rectangles, can be considered as one rectangle.

It's unchanged, the area, because all I've done is moved it, I haven't made it overlap, there isn't any additional area.

I've just moved it to addition a different place.

So that's the rearranging part.

And so this area is 52 square metres because my one rectangle has a length of 13 and a width of four.

With a little bit of thought, some complex compound shapes may also rearrange to simplify the calculations.

So here is our complicated compound shape.

We know that it is and horizontally symmetrical and so I can take a triangle and move it and that sort of completes into a rectangle.

I can take the other one and move it.

And so I then get this compound shape.

This compound shape is exactly the same area as the original one because I've just rearranged the area.

Now I have two rectangles, A and B.

So area of A would be 39 by 10, area of B would be 24 by 30.

And when we find the total area that is 1,110 square centimetres.

So with a little bit of thought and because of the congruency within those triangles, I can rearrange them and make them into rectangles.

So here is a check, this compound shape is made from rectangles.

It has been rearranged into one rectangle, add the dimensions.

So imagine where I may have done some rearranging of the original compound shape to make it just one single rectangle.

And now what would the dimensions be? Pause the video and then when you're ready to check press play.

So it would be a nine, which is the width of the original compound shape by seven because that rectangle that sort of stuck at the top is four by five.

And so I can cut that off there and fit it in in the gap, which is four by five.

So we're on to task C.

So question one, by rearranging the area, find the area of these compound shapes, there are two parts.

So I want you to think about where you could chop and move a part of the area in order to simplify the calculation.

Press pause whilst you're doing that and then press play to move on.

Question two, I want you to find the area of these compound shapes.

This time I'm gonna leave it up to you.

I want you to think about the most efficient method out of the three for the shape that you are working on.

It might be subdividing, it might be completing the shape or it might be rearranging the area that we've just had a look at.

So you need to think about the most efficient method for the composite shape that you are faced with.

Press pause whilst you work through.

And then when you press play, we're gonna move on to two more parts of question two.

So here are two more parts under question two.

Still find the area of these compound shapes.

They're getting a bit more complicated.

So you may need to think about these again, all the different methods that we've met.

But also maybe you need to think about other ways that you could find the area.

Press pause whilst you're working through those.

You can use a calculator.

It's about working out the method as opposed to doing the number work, press play when you're to check your answers.

Okay, so question one, you had to do this by rearranging the area and there was a rectangle that has been removed on part A and filled the gap and that made a rectangle that was then eight by seven.

So it was 56 square metres, that was more efficient than subdividing the shape or completing the shape.

Part B was a bit like a puzzle piece that that octagon was stuck at the other end would fill the gap that was missing.

So it's just a case of chopping on that side and moving it to the other side, we then have a complete rectangle, which is 12 by 10.

So 120 square centimetres.

So we're gonna go through question two, then remember there are four parts to question two.

I'm gonna go through each part.

I've gone with the method I think is the most efficient, but ultimately, check the final answer.

Did you get the final answer correct? Even if you chose a different method.

But then please reflect on, was the method you chose more efficient or maybe you should have considered this method instead.

So this compound shape, I made the decision to divide it into basic shapes.

So I've got a triangle, a rectangle and another rectangle and I'm moving down the shape.

So the area of the triangle was the base times perpendicular height divided by two, which gave you 12.

The area of rectangle one, which is the smaller one was 18 and the area of rectangle two, which the one at the bottom was 120.

So if I add those together, I get 150 square metres.

Part B, I've decided to rearrange the shape.

So there was a triangle section that could literally fit the gap of this triangle.

And then I have a rectangle which is 16 by 20, so that's 320 square millimetres.

So rearranging the shape was what I found most efficient for finding the area.

Moving on to C.

So C, you could have done a different way.

You may have split it up into parts, you may have completed the area just at the top at the 7.

3 level.

I have done this using a slightly different method, which is again, thinking about all parts of maths that we have and how it might be useful.

So due to the rotational nature of the shape, using two copies and then dividing by two is a really efficient way of getting the area because I now know that that rectangle is 3.

8 wide by 7.

Add the four because I've got two copies and then I divide it by two to get just the single area.

So you might wanna pause the video and work through that one again, but do check your area.

21.

47 was the correct answer.

There was a way of doing this by splitting it up, but it definitely was not as efficient as part D, lots going on within this diagram.

So you could use Pythagoras theorem to calculate some of the internal dimensions.

From the given information, you knew that it was a right angle triangle, so we could get that that length of the triangle was 12, which was also a length of the parallelogram.

And then the area can be calculated by subdividing or rearranging.

So you could just keep it as two triangles and a parallelogram.

But you could also move one of the triangles to make a rectangle at the bottom.

So if I look at it as just two triangles and a parallelogram, then the area is 144.

5 square centimetres to one decimal place.

So we can see that there's a seven point zero.

I'm gonna use the most exact form on the calculator within my calculations.

To summarise today's lesson, we've looked at methods to find the area of compound shapes and the three common methods are subdividing the shape, completing the shape and rearranging the shape.

So the most efficient method depends on the features of the shape that you're trying to get the area.

And through practise you'll become more familiar of when to use each of those methods, but there are other methods and combinations of those methods that can also work as well.

Really well done today and I look forward to working with you again in the future.

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