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I can calculate a missing side length from the area of a trapezium.

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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

The formula can be used to calculate the area of a trapezium.
The area of a trapezium can be used to calculate a missing side length.
It cannot be used to calculate one of the slanted sides.

Common misconception

Pupils may forget to divide by 2 when calculating the area of a trapezium.

The area of a trapezium is half of the area of the parallelogram formed from two identical trapeziums.

Keywords

Trapezium - A trapezium is a quadrilateral with exactly one pair of parallel opposite sides.
Trapezia/trapeziums - The plural of trapezium can be referred to as trapezia or trapeziums.

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

By the end of today's lesson, you'll be able to calculate a missing side length if you know the area of a trapezium.

Our lesson has two parts to it and we're gonna start with part one which is calculating the area of a trapezium.

In the previous lesson, we looked at how you can do this via composite shape knowledge and then at the end of the lesson, we did an investigation to derive the formula for the area of any trapezium.

What we didn't do though was actually practise using that formula so that's what we're going to do now.

The formula for calculating the area of a trapezium is the sum of the two parallel sides.

That's a plus b.

The two sides that are parallel to each other multiplied by the perpendicular height and then we divide by two.

Remember, we're dividing by two because in order to derive this formula, we placed two identical trapezium adjacent to each other and showed that they made a parallelogram but because we only want the area of one of those two trapezium, we have to halve.

Here is an example of a trapezium.

I've left the formula up so you can see what it is that I'm referring to.

Let's start by checking that we understand which of the sides here are the ones that are parallel to each other and which of the sides is the perpendicular height or distance between them.

Take a moment, reflect on the diagram and see if you can work out which two measurements are parallel to each other and which measurement represents the perpendicular height.

Let's see if you got it right.

The area of this trapezium is found by adding together the two parallel sides.

That's the two and the nine.

I would then multiply this result by four because four is the perpendicular distance between them or the perpendicular height.

I then divide by two.

Remember, brackets.

This tells us the priority of the operations.

In other words, we must sum the two and the nine together first.

This gives us 11.

We then multiply by four to give us 44 and then we halve that it gives us an area of 22.

Now, if you've correctly typed this into your calculator, your calculator understands those brackets and will do the priority of operations correctly.

So if you type in the very top line of working into your calculator and you're careful, make sure that you've got it all in the right order, your scientific calculator will correctly produce the answer of 22.

Feel free to pause the video and try this.

It's now your turn.

You have a trapezium here on the right with an awful lot of measurements shown.

Identify the parallel sides and the perpendicular height and then use these in the formula to get the area of this trapezium.

Pause the video while you do this now.

Welcome back.

Let's see how you got on.

Did you correctly spot that the two parallel sides have length three and nine and that the perpendicular distance between them is also three? So your calculation should read that you add three and nine together to give you 12.

Multiply this result by three to give 36 and then divide by two to give a total area for the trapezium of 18.

It's now time for your first task.

For each trapezium, please calculate its area.

In order to help you, I've left the formula for the area of the trapezium at the bottom of the screen.

Remember, you're very welcome to use a calculator here.

What I'd recommend though is for any of the values where you feel comfortable doing the calculation, have a go at doing it first on paper and then use the calculator to check your answer.

This is a great way of using that tool in order to check your work and to see if you get to the same result.

If you don't, you've either made a mistake in your working or you've made a mistake using your calculator.

Of course, if you've identified the wrong lengths, the calculator can't spot that but it is good at finding out whether or not you've made a mistake with one of those basic calculations.

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

Welcome back.

Let's see how you got on.

For the first trapezium, we needed to identify the parallel sides and that's the three and the seven.

We add these together to make 10 and then multiply by the perpendicular distance between them which is two.

10 lots of two is 20.

Divided by two gets us to the area of 10.

Let's now look at B.

In B, the two parallel sides have length five and two and the distance that's perpendicular to both of these sides is the two.

Two add five is seven and seven multiplied by two is 14.

14 divided by two is seven.

In C, there was an awful lot going on here.

The two parallel sides and their perpendicular distance between them must be 4.

8 and six respectively with the perpendicular height being 1.

8 plus six is 10.

Multiplied by 1.

3 is 14.

04.

We then have to divide by two giving us an area of 7.

02.

I don't know about you but I used my calculator here.

Although I'm comfortable multiplying the decimals, I wanted a bit of speed and since this is a basic calculation in the sense of it's just a bit of multiplication, I wanted to make sure that I'd definitely done this right.

So the calculator was a useful tool here to make sure my multiplying was correct.

It's now time for the final section of our lesson and this is on finding a missing side length using the area.

Now, we've done this sort of thing before with a triangle.

We had its area and we had to calculate a missing side length.

We did it very recently with the area of composite rectilinear shapes where I gave you that area and then you had to work out what one of the missing side lengths was.

So it's the same thing only we're dealing with trapeziums now.

Given the area of a trapezium, it may be possible to calculate a missing side length.

Let's just check.

True or false, given the area of a trapezium, you can calculate any missing side length.

Do you think that's true or false? Don't forget, when you make your selection, you also need to justify your answer.

So which one of A, the formula for the area of a trapezium involves all the sides of the trapezium or B, the formula for the area of the trapezium only involves the parallel sides and the perpendicular distance between them? Which one of those two supports your choice of saying the statement is true or saying the statement is false? Pause the video now while you make your selection.

Welcome back.

Let's see what you put.

This is definitely false.

The formula for the area of a trapezium only involves the parallel sides and the perpendicular distance between them.

So if I wanted to know what one of the slanted lengths was, I can't use the area of a trapezium to calculate it because that length does not appear in my formula so it's not gonna help me work it out.

There is something though that I could use if I wanted to know a missing slanted length.

What do we use to calculate around a shape? That's right, it's perimeter.

So if I had the perimeter and I wanted to work out one of the missing side lengths, that might be possible but it's not going to be possible with area.

Let's have a practise at this.

I have a trapezium.

I can see that three of the lengths are marked and one is unknown and I'm told that the area is 22.

Just like before, I need to identify which of these lengths here are my two parallel sides, my a and my b, and what is the perpendicular height or distance between these two sides? Can you see that the x and the nine are my two parallel sides? So in my formula, they are my a and my b.

So I know that the area, 22, is equal to the two parallel sides added together so x plus nine then multiply by four and divide by two.

Okay, so the area of one trapezium I found by dividing by two so I can double this and think about it when it was a parallelogram.

So if I double this, 44 must be equal to x plus nine, multiply by four.

I know my four times table.

In order to get to 44 when multiplying by four, I must have multiplied by 11.

So 11 must be equal to x plus nine.

What do you add to nine to make 11? We add two.

So my missing side length must be two units long.

On the right is your trapezium.

You can see that I've marked a lot of different measurements on but there's one that's missing.

I've told you the area is 18.

When you have your turn, remember, identify which of the sides you can see are the two parallel sides and identify which length there is the perpendicular height between those two parallel sides.

Once you've identified that, you can put them into your formula and work out the length of the missing side.

Pause the video and do this now.

Welcome back.

How did you get on? Let's first of all check that you selected the correct sides and put them into the formula.

The area is 18 and that's equal to the sum of the two parallel sides so that's three, add nine then multiply by the perpendicular distance between them.

On my diagram, that's marked as y, the length that I'm trying to find.

Let's tie that up a bit.

Three plus nine is 12 so that's where the twelves come from.

Now, rather than halving this, I'm going to double my area.

So 36 must be equal to 12 lots of y.

What do we multiply 12 by to get to 36? That's right, it's three.

In other words, y has a length of three units.

It's now time for our second and in fact, final task.

I'd like you to calculate the length of the marked side and then if possible, calculate the perimeter of the trapezium as well.

Remember, it's not always possible but if I've already given you enough information then you will be able to calculate the perimeter.

The area for each trapezium appears underneath it.

So for A, the area of the trapezium is 22.

For B, the area is 10,and for C, the area is 32.

64.

Pause the video now while you calculate the length of each marked side for each respective diagram and then if possible, calculate the perimeter of each trapezium.

Welcome back.

Let's see how you got on.

Let's start with A.

The two parallel sides have length eight and an unknown length for b.

We know that the perpendicular distance between them is four and the area is 22 so we can write our formula putting in the information we know.

22 is equal to equal b added to eight then multiply by four and divide by two.

Well, that tells us if we double, that 44 is equal to b plus eight multiplied by four.

We know what 44 divided by four is.

It's 11 so b plus eight must be equal to 11.

What do you add to eight to make 11? You add three.

So that means that b has a length of three metres.

Now, you might think we have one missing side length, the other slanted sides so we can't do it but actually, we can see the notation that tells us the two slanted lengths are the same length which means that the five metres I can see is the same for both of them so we can calculate the perimeter.

It would be eight plus five plus five plus what we got for b.

In this case, we know b is three.

So eight plus five plus five plus three giving us a total perimeter of 21.

Let's now look at B.

In B, we can see that the two parallel sides are the lengths of three and seven so we add those together and multiply by the perpendicular distance between them which is d.

Three plus seven is 10.

So 10 multiplied by d divided by two is 10.

Well, we can double that.

So 20 is equal to 10 multiplied by d.

What do you multiply 10 by to make 20? It's two.

So d must stand for two.

In other words, that missing length is two centimetres long.

Can we calculate the perimeter? Well, yes, we can.

We know all of the sides.

So seven, add four, add three, add the two that we just worked out gives us a total of 16 centimetres.

Let's now look at C.

In C, we can identify that the two parallel sides are the unknown one marked h and the length of 12 so h plus 12 then multiply by the perpendicular distance between them which is 3.

4 and divide by two.

This gives us our area of 32.

64.

Well, we can double that.

So 65.

28 must be equal to the numerator of our fraction so h plus 12 multiplied by 3.

Now, at this point, I used my calculator.

65.

28 divided by 3.

4 is 19.

2 so that must be what h plus 12 is equal to.

What do we add to 12 to make 19.

2? We add on 7.

2 and that must be what h stands for.

So that unknown length is 7.

2 centimetres.

Can I calculate the perimeter? I can't because one of the sides of my trapezium is still unknown and there's nothing to indicate that it's equal to any of the other sides that I know so I cannot calculate the perimeter in this instance.

Remember, just because I could use the area to find a missing side length doesn't mean I'll have all the information I need to calculate the perimeter because remember, the formula for the area of a trapezium involves the two parallel sides and the perpendicular distance between them.

It doesn't involve any of those slanted side lengths.

Let's summarise what we've done today in our lesson.

The formula for calculating the area of a trapezium is the sum of the two parallel sides multiplied by the perpendicular distance between them and then divided by two.

You could also say that it's the sum of the two parallel sides multiplied by the perpendicular height and then multiplied by a half because remember, multiplying by a half is the same as dividing by two.

This formula can be used to find the length of either one of the parallel sides or the perpendicular distance between them.

What it can't be used for is to find the length of one of the slanted non-parallel sides.

That isn't possible because that length does not appear in my trapezium so I can't calculate one of those lengths using this formula.

It's worth bearing that in mind.

Of course, it doesn't mean that you can't find the perimeter of the trapezium.

If that information i.

the length of those slanted slides was already given to you then the formula could be used to find one of the missing parallel sides and you would be able to calculate the perimeter.

Well done.

You've worked really well today and I look forward to seeing you in our very final lesson for this unit where we'll be looking at problem solving with perimeter and area.

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" } ], "id": "6002af50-7013-3622-de82-3235b78b8f54", "lessonId": "1cc7e459-ea5d-16bf-eda1-471dea675933" }, { "type": "Question", "question": { "text": "Calculate the area of the trapezium (you do not need to state the units).", "image": "https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1690273941%2Fujn5q0ik4eve5s064uvc.png&w=640&q=75", "imageKey": "8228ac47-1ad4-c44c-77cf-ef21ca07947e/question/53d16c0a054a92d26084d6e863cd4194.png" }, "questionType": "Short answer", "correct": [ { "text": "13.125" }, { "text": "13.125 cm

^2

" } ], "id": "2d528781-8c76-6660-a707-b1dc8d58eeff", "lessonId": "1cc7e459-ea5d-16bf-eda1-471dea675933" }, { "type": "Question", "question": { "text": "Calculate the area of the trapezium (you do not need to state the units).", "image": "https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1690273941%2Fvt9iiimxuqwrnlpptjxp.png&w=640&q=75", "imageKey": "8228ac47-1ad4-c44c-77cf-ef21ca07947e/question/e308c35295fbf15dbe15e22a3f4d7975.png" }, "questionType": "Short answer", "correct": [ { "text": "9.35" }, { "text": "9.35 cm

^2

" } ], "id": "f2a499ab-6331-3c1a-ef13-6d159e17d575", "lessonId": "1cc7e459-ea5d-16bf-eda1-471dea675933" }, { "type": "Question", "question": { "text": "What is the value of

b

?", "image": "https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1690273941%2Ft5ektefw3sictww4ml33.png&w=640&q=75", "imageKey": "8228ac47-1ad4-c44c-77cf-ef21ca07947e/question/aa9d9a7d817d3b8847630cb48949812d.png" }, "questionType": "Multiple choice", "correct": [ { "text": "4" } ], "choices": [ { "text": "4" }, { "text": "14" }, { "text": "16" }, { "text": "32" } ], "id": "e80400a8-b5c5-7afc-f127-bd26fc22c335", "lessonId": "1cc7e459-ea5d-16bf-eda1-471dea675933" }, { "type": "Question", "question": { "text": "What is the value of

d

?", "image": "https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1690273941%2Fhnhewmcccre0a1o0po4j.png&w=640&q=75", "imageKey": "8228ac47-1ad4-c44c-77cf-ef21ca07947e/question/6c8f5967f830e8e9c8725c5047d43996.png" }, "questionType": "Multiple choice", "correct": [ { "text": "7" } ], "choices": [ { "text": "5" }, { "text": "5.5" }, { "text": "6" }, { "text": "7" } ], "id": "17b341cc-13f9-9254-510e-ff6a8c26e901", "lessonId": "1cc7e459-ea5d-16bf-eda1-471dea675933" }, { "type": "Question", "question": { "text": "What is the value of

h

?", "image": "https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1690273942%2Fonmglzhsjc5fkxvnn5m3.png&w=640&q=75", "imageKey": "8228ac47-1ad4-c44c-77cf-ef21ca07947e/question/f02f5d3eff293e3be5f51f2e271f92e2.png" }, "questionType": "Multiple choice", "correct": [ { "text": "11" } ], "choices": [ { "text": "3" }, { "text": "5" }, { "text": "9" }, { "text": "11" } ], "id": "c2cc6263-4ad6-b0cc-24b6-93db9c995533", "lessonId": "1cc7e459-ea5d-16bf-eda1-471dea675933" } ], "lessonId": "1cc7e459-ea5d-16bf-eda1-471dea675933", "meta": { "gradeStageId": "ks3", "subject": "Maths", "grade": "Year 7", "topic": null, "unit": "Perimeter and area", "lessonCount": "12", "age": [ 11, 12 ], "gradeId": "age_11_12", "gradeStage": "Key Stage 3", "lesson": "Calculating missing side lengths from the area of a trapezium" } } ] }