Lesson video

Hi, welcome to today's lesson on the area of a trapezium.

By the end of today's lesson, you will have derived and be able to use the area of a trapezium formula.

Before we get started, we're just gonna review a couple of key words, and that's the word trapezium and the words trapezia or trapeziums. Now, you may find that it's easier for you to do some drawing here because I'd like you to write down what your understanding of these words is.

If you find it hard to describe but would rather draw some pictures, you're very welcome to.

Pause the video now while you write down what these words mean.

Welcome back.

Let's see if your definitions match mine.

A trapezium has one pair of parallel opposite sides.

It's also a type of quadrilateral.

The plural of trapezium can be referred to as trapezia or trapeziums. In other words, if you have more than one, there are two different ways to refer to it, and you may see either word used interchangeably.

This is just so you're aware and that you don't think it's only one or the other.

This is our lesson today.

We're starting with Section 1 on building an area.

Hmm, this sounds like something we've done before.

When we were looking at the area of composite rectilinear shapes, we built an area by putting different rectangles together and that created our composite rectilinear shape.

Perhaps this section has something to do with that.

Let's see.

The areas of basic shapes can be used to create a new composite shape with a given area.

So this time, we're not just talking about composite rectilinear shapes, I.

e.

, ones built with just rectangles.

We're talking about composite shapes.

Remember, a composite shape is a shape built from other basic shapes placed together.

So, for example, here, if I have a square with an area of 90 and a triangle with an area of 35, combining these two shapes, making sure that they're adjacent, not overlapping, would give a total area of 125 'cause it's the area of the square plus the area of the triangle giving us the area of the composite shape.

Do any of these composite shapes have different areas? You're right, they don't.

What I've done is taken the rectangle and the square and placed them next to each other.

Because they are not overlapping in any of these diagrams, all of them must have the same area.

Now, a little bit of a puzzle for you to solve.

Which of the shapes that you can see on the page could be combined to create a new shape with an area of 200? Could I combine two triangles to make an area of 200? Could I combine one of the squares and one of the triangles to make an area of 200? Or could I combine a square, a triangle and a parallelogram to make 200? Select the options that you think are correct.

Pause the video now while you do this.

Welcome back.

Did you spot that there's more than one correct answer here.

That's right.

If I combine two of the triangles, I'll get a total area of 200 because 100 plus 100 is 200.

If I combine a square, a triangle, and a parallelogram, that's 25, add 100, add 75, which also makes 200.

The only one that doesn't work is the square in the triangle because that only gives an area of 125.

It's now time for our first task and you're going to build me some areas.

What I'd like you to do is state how many of each shape should be used to create a composite shape with the area stated below? In other words, for part A, I want a composite shape with an area of 24.

Which of the shapes above should I use to make that area? And how much of each of the shapes should I use? For some of them, there is more than one way to do this.

It doesn't matter which one you pick, just make sure that the total area is 24.

If you like, you can even have a go at drawing the composite shape to show what you used.

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

Welcome back.

Let's see how you got on.

For A, there were multiple different things you could do.

You could have said you want to use three triangles because three lots of eight is 24.

You might have said you want to use one square and one triangle because the area there would be 16 add 8, which is also 24.

Remember, different ways to get the same answer.

In B, you had to get to 50, and I've suggested that you can do this by using two of the squares and a parallelogram because 16 add 16 add 18 is 50.

You could, of course, have used two triangles in place of one square.

So potentially you said four triangles and one parallelogram.

That would've worked as well.

And then in C, you had to get to 60.

I suggested using one square, one triangle and two parallelograms. But again, there are other things you could have used too.

The important bit is to check that your calculation is equivalent to 60.

It's now time for our second section.

We're going to be using what we've covered in section one, I.

e, building an area to consider how we could calculate the area of a trapezium using this composite shape knowledge.

Let's have a look at it now.

Any trapezium can be made using a rectangle and a triangle, and we can see that here.

I've got a rectangle, I have a triangle.

I've cut my triangle in half and I've stuck it on either side of my rectangle to form a trapezium.

The area of a trapezium is therefore equal to the area of its rectangle and the area of its triangle because it's a composite shape.

You can see on the diagram on the left, my trapezium, that the bottom of my trapezium has a total length of eight.

And if I look at the two lengths on the bottom of my two basic shapes, I have a length of five and a length of three, making a total of eight.

The top of my trapezium has a length of five, as does the top length of my rectangle.

And then we can see the perpendicular height is marked as two, which is both the height of the rectangle and the height of the triangle.

So the area of my trapezium can be found by calculating the area of my rectangle, so two multiplied by five, add to the area of my triangle.

Remember, the area of the triangle is half of the area of the parallelogram you can form.

So it's the base times the perpendicular height and then either multiplied by a half or divide by two, whichever you find easiest to remember.

In other words, two multiplied by five is 10, and two lots of three is six.

Half of six is three.

So the area of the trapezium is 13.

So what we did was we took a composite shape, one that initially looked quite complicated, broke it down into the two basic shapes that made it and found the area.

Let's have a look at this trapezium.

Remember, I want the area of the rectangle added to the area of the triangle.

Well, the area of the rectangle was the top height of my trapezium multiplied by the perpendicular height.

So three multiplied by seven.

To find the base of my triangle, I had to know the difference between the top and the bottom measurement.

So 11 takeaway three gives me what the base of my triangle would be.

I then multiply by the perpendicular height and halve this.

So let's tidy up my calculation.

Three lots of seven is 21.

11 takeaway three is eight multiplied by the perpendicular height of seven multiplied by 1/2.

So that'd be 21, and I did eight times 1/2 is four, so I just had to work out four times seven.

You could have done eight times seven is 56 and then halved it.

Entirely up to you the order you work this out in remember because this is multiplication.

So 21 plus four lots of seven means 21 plus 28 or an area of 49.

It's now your turn.

You have a trapezium on the right of the screen.

Work out its area.

Pause the video now while you do this.

Welcome back.

Let's see how you got on.

Remember to start by looking at the area of the rectangle.

It's a two multiplied by six.

Then we need the base of the triangle which is found by doing 10 subtract two, giving us a base of eight, multiplied by the perpendicular height of six and then halved.

Let's tidy up our calculation.

Six times two is 12.

10 takeaway two is eight, and half of six is three.

So we have 12 plus eight lots of three, or in other words, 12 plus 24 for a total area of 36.

It's now time for our second task.

I'd like you to calculate the area of each trapezium using our composite shape method that we've just been practising.

Check carefully for which measurements you're using 'cause these trapezium looks slightly different to the ones we've already seen, but it still works the same way.

Pause the video and have a go at this task now.

Welcome back.

Let's see how you got on.

For A, we can see that the rectangle is 17 times nine.

We are then left with a triangle on the end, and in fact, it's a right angle triangle.

This doesn't change how we calculate its area, however.

To find the base of that triangle, we find 20 subtract nine, which will give us a base of 11 multiplied by the perpendicular height of 17 and then halved.

So we've then tidied up that calculation.

So nine times 17 is 153, and we have 11 times 17 and then halved.

Well, 11 times 17 and then halved is 93.

5.

I don't know about you, but I used a calculator for this bit.

I then worked out 153 plus 93.

5 giving us an area of 246.

5.

Let's now look at B.

In B, I saw that the area of the rectangle would be five times four.

Then to calculate the base of the triangle, I did 12 subtract five multiplied by the perpendicular height of four and then halved.

Well, five times four is 20, and then I'm adding 12 take away five, which is seven, multiplied by four and then halved.

Well, seven times four is 28.

This halved is 14.

So 20 add 14 gives me an area for b of 34.

It's now time for the final section of this lesson.

We've worked out that we can calculate the area of trapezium via our composite shape knowledge, but is there a more formal way? Well, actually, there is, and we can deduce the formula for the area of a trapezium via an investigation.

Does that sound familiar? We did an investigation before to derive a formula for a shape.

Can you remember what shape it was? That's right, it was the triangle.

So similar idea here.

Let's investigate.

We'll start with a quick check.

Is the shape that you can see here on the screen a trapezium? If you think it is, you pick true, and if you think it's not, you pick false.

Remember though, you need to justify your answer.

So which of A or B justifies what it is you're choosing? Pause the video now while you make your selection.

Welcome back.

What did you go for? This shape is a trapezium.

Remember the start of our lesson where we had our definition of a trapezium? We said it was a quadrilateral with one pair of parallel sides.

Well, let's look.

That is a quadrilateral, and it has one pair of parallel sides, so it must be a trapezium.

A trapezium is indeed made of a rectangle and triangle, but that's not helping us to justify why this shape here is a trapezium.

Does this animation help? It's our trapezium from before.

It's just showing how you can make that rectangle and triangle.

The base of the triangle comes from both parallel lines.

In other words, the method that we have been using is not going to work here.

So we need a better method.

Remember, our previous method isn't working because we can't just subtract the top length from the bottom length because the top length, can you see, included part of the base of the triangle.

It's time for our investigation.

Now these instructions will look familiar.

I want you to start by taking a piece of paper.

Last time, I suggested A4 was best but really as long as you can draw and cut out with it, it doesn't matter how big the paper is.

Take that paper and fold it in half.

When you've done that, you're ready for the second instruction.

Draw a trapezium on your piece of folded paper.

Do try to make it different from mine because you're going to see what mine looks like, and you want to investigate and try something different.

Just make sure, though, it is a trapezium.

When you've done that, you're ready for the next instruction.

Cut out your trapezium.

You should now have two identical trapeziums because of your folded paper.

Very similar to when we did the area of a triangle.

Once you're done, you're ready for the next instruction.

I'd like you to rotate one of the trapezium by half a turn.

In other words, turn it upside down.

Place the two trapezium next to each other.

Which quadrilateral have you made? You have made a parallelogram.

Well, now that's odd, isn't it? When we did the area of a triangle and we placed our two identical triangles adjacent to each other, we could always make a parallelogram.

Well, now we've done it with two trapezium, we've made a parallelogram again.

The area of a trapezium is therefore half the area of the parallelogram it makes because two identical trapezium can always be placed together to create a parallelogram.

Hang on a second.

Does this mean that the formula for finding the area of a trapezium is going to be exactly the same as the formula for finding the area of a triangle? After all, the area of a triangle is half of the area of a parallelogram, and the area of a trapezium is half the area of a parallelogram.

Surely that means the two things are the same.

Does it though? Here on the left, we have a parallelogram, and on the right we have the parallelogram formed by having two trapeziums together.

Let's just take a moment to see what's similar between the two and what's different.

Well, that's interesting.

Can you see that in the shape on the left, the base of the parallelogram from our two triangles goes all the way across? But in the parallelogram formed by putting two trapezium together, I have the bottom side from one trapezium, but because I turned the other one upside down, I've also got the top length of the trapezium.

So in fact, to make that parallelogram, I don't just have the base of my original shape or just one side of my original shape, I have both the bottom side of the original shape and the top side.

In other words, it's not just one length of my original shape that forms the base of the parallelogram.

Ah.

I can now start to see why the formula for this is going to be different.

Let's just recap how to find the area of a parallelogram.

The area of parallelogram is the base times its perpendicular height.

So using the notation I can see on the left, that would be b multiplied by h.

Now let's look at the parallelogram on the right, the one that's been formed from my two trapeziums. The area of one of the trapeziums will be the base of the parallelogram multiplied by the perpendicular height of the parallelogram and then halved because I only want one trapezium's area, not the area of two of them together.

Well, what's the base of that parallelogram? I can see the base of the parallelogram is the length b added to the length a.

So b plus a is the entire base of my parallelogram.

I then multiplied by the perpendicular height of the parallelogram, which is h, and remember, I'm halving it.

So in other words, it's very similar to the formula for the area of a triangle.

The only difference being that when I used two triangles to make a parallelogram, the base of my triangle was the entire base of the parallelogram.

But when I used two trapeziums, the base of the whole parallelogram was found by adding together the two parallel sides from my original trapezium.

Let's do a quick check.

Is the statement, the formula for the area of a trapezium is the same as the formula for the area of a triangle.

That always true? Sometimes true? Or never true? Pause the video while you make your selection.

It's never true.

For the formula of the area of the trapezium to be the same as the formula for the area of the triangle, one of my parallel sides of my trapezium would need to have a length of zero.

Well, hang on a second.

If that was true, then my trapezium is not a trapezium because it's not a quadrilateral anymore, it is, in fact, become a triangle.

Let's summarise everything we've learnt today.

A trapezium can be thought of as a composite shape made from a rectangle and a triangle.

The area of any trapezium is half the area of the parallelogram made from two copies of that same trapezium.

And the formula for the area of a trapezium can be written in many different ways.

On the screen, you can see three equivalent ways to say the same thing.

All three of these formula say that we should add the two parallel sides together to give us the base of the parallelogram they form multiplied by the perpendicular height and then halved.

Well done.

You've worked really well today.

I look forward to seeing you in our next lesson.