Lesson video

Hi there, my name is Mr. Tilstone.

I'm a teacher.

My favourite subject is definitely, definitely maths.

So it's a real honour to be here with you today to teach you this lesson, which is going to be about the area of a triangle.

You might have had some recent experience finding the area of a triangle.

Let's see if we can take that one step further today by looking at missing measurements on a triangle.

If you're ready, I'm ready.

Let's begin.

The outcome of today's lesson is I can use the area of a triangle to calculate unknown measurements, and our keywords today are my turn, base, your turn.

And my turn, perpendicular, your turn.

I suspect you've heard those words quite a lot recently, but let's have a little reminder of what they mean because they're very important.

You'll need to know these today before we begin.

The base is the side which is perpendicular to the shape's height, and two lines are perpendicular if they meet at a right angle.

So you might recently have been using the base and the perpendicular height and multiply them together to get a parallelogram and then halving that to get the area of a triangle.

Our lesson is split into two cycles.

The first will be, use division to find unknown height or base, and the second, drawing triangles with a known area.

So if you are ready, let's begin by thinking about how we can use division to find unknown heights or bases.

In this lesson, you're going to meet Jun and you're going to meet Alex.

I'll bet you've met them before.

They're here today to give us a helping hand with our maths.

This parallelogram has a missing dimension.

Can you spot what it is? Well, let's see what it has got.

The area is mentioned, that's 15 centimetres squared.

The base is given, that's five centimetres.

So the missing dimension is its perpendicular height.

I wonder how we might calculate it.

How can we work it out? How can we figure it out? Well, can you remember this formula? Base multiplied by perpendicular height equals area.

Remember using that? I think we can still use that here because we can see the base is five centimetres multiplied by something.

We don't know what that is yet.

The perpendicular height, we don't know, but we do know the area that's 15 centimetres squared.

So five times something equals 15.

That's like a times tables factor, isn't it? Or we could think of the inverse division.

We could think of it as area which we know divided by base, which we also know equals perpendicular height.

We've got those two pieces of information to work that one out.

Either way we'll get the same answer.

We'll arrive at that perpendicular height.

So that will be 15 centimetres squared divided by five centimetres equals something centimetres.

All triangles have an area half of a parallelogram with the same perpendicular height and base.

And that might be something that you've explored recently.

Does that ring any bells? A formula can be used to calculate the area of a triangle.

And I'm going to show you that triangle in a second.

But before I do, do you think you can remember the formula or work it out? There's a clue.

Area equals base multiplied by perpendicular height divided by two.

Look at these two triangles.

What's the same and what's different about these examples? Have a look carefully because on first glance they look very similar, don't they? But there's some different information missing on each one and different information given on each one too.

Let's have a look.

They are the same shape.

You might have noticed that, they're both right angled triangles.

Now, in the first example, the area is unknown.

In the second example, the area is known, but the perpendicular height is unknown.

What formula could be used to calculate the missing area? Can you remember? Can you work it out? Base multiplied by perpendicular height divided by two equals area.

So in this case, 10 centimetres multiplied by eight centimetres divided by two equals something centimetres.

And in the other triangle, what formula could be used to calculate the missing perpendicular height? Think about how we could use the information we do have to find out the information we don't have.

Let's have a look, while multiplying the triangle area by two gives the area of the parallelogram that could be formed by combining two congruent triangles.

So if we add that triangle, again, an exact replica of it, we could form a rectangle in this case or a parallelogram, you might want to think of it as just like this.

So that's area multiplied by two.

Now, dividing the area of the parallelogram by the known measurement, in this case that's a 10 centimetre base, will then give the unknown perpendicular height.

So area multiplied by two divided by base will give the perpendicular height so that formula will work in this case when the perpendicular height is missing.

So in this case, to put some numbers to that, that's 40 centimetres squared multiplied by two divided by 10 centimetres, that's the base, equals something centimetres.

That will be the perpendicular height.

Area multiplied by two divided by base equals perpendicular height.

When the area of the parallelogram is known, the final step can be written as a missing multiplication fact.

So here we go.

That will be the complete rectangle.

40 centimetres squared, that's the area of that triangle multiplied by two.

That will give you the rectangle equals 80 centimetres squared and then 80 centimetres squared divided by 10 centimetres equals something centimetres.

But to think of that as a multiplication fact with a missing part to it, we could think of it like this.

10 centimetres multiplied by something equals 80 centimetres, and either way we get the answer, eight.

So the perpendicular height is eight centimetres.

Let's have a check for understanding.

Let's see how much of that you've taken on board.

Use the formula and it's there.

If you need a reminder, it's there to calculate the missing perpendicular height.

Pause the video and have a go.

(no audio) Well, now we've got the area, haven't we? And we've got the base so we can double that area and multiply it by two to give us what would be the rectangle and then we can divide it by the base that's 12 centimetres to give us that missing perpendicular height.

So that's 60 centimetres squared multiplied by two equals 120 centimetres squared, and then 120 centimetres squared divided by 12 centimetres equals something.

Or you could think of it as that missing multiplication fact.

That's 12 centimetres multiplied by something equals 120 centimetres.

And either way the answer is 10 centimetres.

So big congratulations if you've got that, you are definitely on track.

The formula can be used to calculate the perpendicular height of all triangles, not just right angle triangles, because remember, any pair of congruent triangles can form a parallelogram.

So area multiplied by two divided by base equals perpendicular height.

So here look, we've got our area that's 44 centimetres squared.

So we multiply that by two and then we divide the answer by 11 to give us the missing perpendicular height.

So that is the parallelogram or one of the two parallelograms that could be formed by adding another congruent triangle.

So that's 44 centimetres squared multiplied by two equals 88 centimetres squared.

Then 88 centimetres squared divided by 11 centimetres equals something or 11 centimetres multiplied by something gives us 88 centimetres.

And either way, the missing perpendicular height is eight centimetres.

So it's not just right angle triangles that that formula works for.

It's any triangle.

Let's have another check.

Calculate the missing perpendicular height.

So you've got the area, you've got the base.

What is the missing perpendicular height? The formula is not there this time.

So let's see if you can remember it or work it out.

Pause the video and have a go.

(no audio) Welcome back.

How did you get on with that? Let's have a look.

So the area multiplied by two divided by the base equals a perpendicular height.

That's our formula.

So that's 27 multiplied by two or double, that's 54 centimetres squared is the area of the parallelogram.

And then 54 centimetres square divided by nine centimetres.

That's a times tables fact, isn't it? Equals something or nine centimetres multiplied by something equals 54 centimetres.

And either way the answer is six centimetres.

So if you've got six centimetres for the missing perpendicular height, very well done.

Let's crack on.

So we've got area multiplied by two divided by base equals perpendicular height.

So 10 centimetres squared multiplied by two divided by five equals something.

Here the base is unknown.

How might the formula be amended? So we've got a different missing piece of information and a different known piece of information as well.

We know this time the perpendicular height.

I think we can do something a bit similar to before.

What do you think? Have you got any ideas? Well, we can think about what the area of the parallelogram would be in this case, a rectangle.

Multiplying the triangle area by two gives the area of the parallelogram that could be formed by combining two congruent triangles.

So here we go, so we know the area just like before.

We can double the area just like before.

And then dividing the area of the parallelogram by the known measurement.

In this case, a four centimetre perpendicular height will then give the unknown base.

So it's almost identical to before, isn't it? So this time though, it's divided by perpendicular height rather than base, but we're still starting by doubling the area.

So area multiplied by two, divided by perpendicular height equals base.

That's our formula this time.

So to put some numbers to that, that's 10 centimetres squared multiplied by two, divided by four equals something centimetres.

And the answer to that is five.

So the missing base is five centimetres, very similar to before, wouldn't you agree? So let's have another check then.

Can you calculate the missing base this time please? You haven't got the base, you have got the area and you have got the perpendicular height.

Pause the video and have a go.

(no audio) How did you get on with that? Let's have a look.

Well, we know the area, don't we? We know the perpendicular height, so we can use those.

We can double the area, multiply it by two, and then we can divide it by that perpendicular height to give us the base.

So we're doubling 20 centimetres squared, that's 40 centimetres squared.

That would be the area of the parallelogram.

And then we're dividing by that known perpendicular height, that's four centimetres, to give us the base.

So 40 centimetres squared divided by four centimetres equals or four centimetres multiplied by something equals 40 centimetres.

But either way, whichever way you like it's 10 centimetres, well done if you got that.

I think it's time to do some practise.

Let's see if you can put those skills to action.

Give them missing perpendicular heights each time for these, different piece of information are given, be careful because some of the information is a red herring.

You've got some side lengths on there that aren't going to be helpful.

So be careful that you're using the right two measurements each time.

And then for number two, you're going to give the missing base.

Good luck with all of that.

If you can work with somebody, I always recommend doing that so that you can bounce ideas off each other and remind each other about strategies and methods and things like that.

Pause the video, good luck and I'll see you shortly for some feedback.

(no audio) Welcome back.

Let's have a look at some answers.

So number one, the missing perpendicular heights are as follows.

So for the first one, we've got the area that's 24 centimetres squared, doubled, multiplied by two equals 48 centimetres squared, and then 48 centimetres squared divided by our known base, that is six centimetres, gives us eight centimetres, is the missing perpendicular height.

And then for the next one, we've got three measurements there, but you only needed two.

So well done, if you ignore the 10 centimetres side length, we didn't need that.

That's not the perpendicular height or the base.

So we've got that area and we're going to double that.

And then we're gonna divide it by the base and that gives us nine centimetres.

And for C, you may have noticed that the base wasn't horizontal or vertical and you might have tilted your page so that it was or you might not have done.

So we know our area and we know our base, so we can double our area and divide by our base.

And that gives us 12 centimetres for the perpendicular height.

The arithmetic there was a bit trickier, wasn't it? It wasn't the times tables fact.

And then for number two, we're giving the missing base.

So in this case we know the area again and we know the perpendicular height.

So that's 84 centimetres squared multiplied by two equals 168 centimetre squared is the area of the parallelogram.

And then that number divided by 14, which is our known perpendicular height, gives us 12 centimetres.

That's our base.

And for B, the missing base is seven millimetres.

And for C the missing base, which you might have noticed was not vertical or horizontal.

That was 110 millimetres.

Well done, if you got those.

Did you notice the mixed units and remember to convert? Did you spot that on C? One was centimetres, one was millimetres.

So well done if you turn that 110 millimetres into 11 centimetres.

I think you are probably ready for the next cycle.

And that's drawing triangles with a known area.

Let's go.

Alex is drawing some triangles with an area of 24 centimetres squared.

Using the area he could calculate some possible missing measurements.

He says, "I'm going to think about the parallelogram, which would be halved to make the triangle.

The area of that would have to be 48 centimetres squared." Yeah, I agree it would, wouldn't it? If he's got a triangle with an area of 24 centimetres squared, and the same one, a congruent one would make 48 centimetres squared.

So that is the parallelogram that could be formed by doubling that triangle.

The base and perpendicular height would have to be a factor pair of 48.

So have you got much experience using factor pairs? Can you remember factor pairs? Let's have a look at that.

Alex uses a table.

Good on you, Alex.

I always recommend using tables for this kind of thing to investigate possible measurements which are factor pairs of 48.

So we've got our base and our perpendicular height both in centimetres, and here's some possible factor pairs.

One times 48 equals 48, 2 times 24 equals 48, 3 times 16 equals 48, 4 times 12 equals 48 and six times eight equals 48.

Did you notice how we're systematic there? Starting with the lowest possibility and working upwards, that's all of our factor pairs to 48.

We could look at them the other way round as well.

So exactly the same numbers, but the base and the perpendicular height have been swapped.

So that's all some different possibilities that he could use for his triangle.

And he's going to choose a factor pair.

It doesn't really matter which.

He's chosen this one.

So he's going to go for a perpendicular height of four centimetres and a base of 12 centimetres.

So he's going to draw that.

So he starts by drawing that base of 12 centimetres with his ruler.

Remember all the rules about drawing a straight line where you start on the ruler and that kind of thing, starting at the zero, being as accurate as possible? So here we go.

He's drawn that 12 centimetre base.

That's not the only place that the base could have gone, it's just one place.

In this case it's a horizontal base.

He says, "I can draw the perpendicular height of four centimetres anywhere along the base." He's right, he could even draw it outside the base if he liked, but he's going to do it just here.

So he's starting again with the zero on the base and he's drawn a four centimetre perpendicular height, moves his ruler away.

The parallelogram would have a base of 12 centimetres and a perpendicular height of four centimetres, so it would have an area of 48 centimetres squared.

The triangles area is half out of the parallelogram.

The triangles area is 24 centimetres squared.

So as soon as he'd drawn that base and that perpendicular height, all he had to do was form a triangle around it.

Could a different triangle be drawn with the same base and perpendicular height? So another base of 12 centimetres and another height of four centimetres, but different to that.

What do you think? What could he do? Would it work with a different base and perpendicular height? So not 12 and four, a different factor pair? What do you think? Well, it could have made a right angle triangle with a base of 12 centimetres and a perpendicular height of four centimetres.

So on the left you can see Alex's first triangle, the one you've just seen.

And on the right, Alex's new triangle.

So they've both got a base of 12 centimetres and they both got a perpendicular height of four centimetres, but they look different.

A different factor pair could have been used.

The area is still 24 centimetres squared.

How about we go for this one? A perpendicular height of six centimetres and a base of eight centimetres.

Here we go.

There's the base.

There's one of the places that a perpendicular height could have gone and all we've gotta do is form a triangle around it.

So that is a triangle with an area of 24 centimetres squared, a different one this time.

Is Jun correct? Jun says to draw a triangle with an area of 20 centimetres squared, I could draw a base of five centimetres and a perpendicular height of four centimetres.

Is he right? What do you think? Pause the video, have a chat with your partner if you can, and see if you agree or disagree and why? (no audio) What do you think? Is Jun correct? He's not, is he? He's kind of halfway there though.

He needs to double the area first.

The parallelogram would have an area of 40 centimetres squared and the base and perpendicular height need to therefore be a factor pair of 40, not 20, such as, for example, five centimetres and eight centimetres.

Maybe you can think of a different factor pair for 40.

Halving that would give a triangle with an area of 20 centimetres squared.

So he would need, for example, to draw something like a base of five centimetres and a perpendicular height of eight centimetres.

Let's do some practise, some final practise.

Number one, this triangle has an area of 48 centimetres squared.

What could the base and perpendicular height be? Number two, draw some triangles with an area of 30 centimetres squared.

So just a bit like what you've just seen with Jun there, this time 30 centimetres squared.

Think about factor pairs to 30 and don't stop at just one because there are absolutely loads and loads that you could draw.

So keep going and explore as many as you possibly can.

Have fun with that and I'll see you shortly for some feedback.

(no audio) Welcome back.

How did you get on drawing those triangles? Let's have a look at some answers.

So number one, if the triangle's got an area of 48 centimetres squared, what could the base and perpendicular height be? Well, doubling 48 gives us 96.

So the surrounding parallelogram has got an area of 96 centimetres squared.

In fact, a pairs of 96 are one times 96, two times 48, three times 32, four times 24, six times 16, and eight times 12.

They're all possible factor pairs to 96.

Of those possibilities, looking at that particular triangle, eight multiplied by 12 looks like the closest fit to the triangle's base and perpendicular height.

So well done than if you've got eight multiplied by 12 and well done, if you've got all of those possibilities as well and draw some triangles with an area of 30 centimetres squared.

So many possibilities for this one, there are so many factor pairs to 60, and here they are, one times 60, two times 30, three times 20, four times 15, five times 12, six times 10.

Let's pick one of those.

Let's go for five centimetres by 12 centimetres.

Here's our 12 centimetre base.

Here's our five centimetre height.

Here's a triangle that we can form around it.

So that's one of many, many, many possibilities.

Here's a different one.

Within each factor pair, there are many possible ways to draw a triangle with the same area.

So again, there's the factor pairs.

This one was five by 12.

Again, this was a right angle triangle.

The base was 12, the perpendicular height was five, but a different version of that.

And here's another five by 12 triangle.

Here's our base of 12 centimetres, here's our perpendicular height of five centimetres.

And you might notice the base wasn't horizontal this time and it wasn't vertical.

So it had in a different orientation.

So very well done to you if you drew one of your triangles or more with a base that wasn't horizontal or vertical.

We've come to the end of the lesson.

Today's lesson has been using the area of a triangle to calculate unknown measurements.

So in all of our examples today, we've known the area where there's been something missing.

So maybe the perpendicular height or maybe the base.

So if the area is known, formulae can be used to calculate the unknown base or perpendicular height.

So here look, we've got an example where we know the area that's 40 centimetres squared and we know the base that's 10 centimetres.

We don't know the perpendicular height.

We can use a formula that's area multiplied by two divided by base equals perpendicular height.

So if the area of a triangle is known but the base and perpendicular height unknown, various possibilities exist, which can be determined by doubling the triangle area to create a parallelogram and then finding factor pairs of the doubled number.

So let's see if we can work out this missing one.

Let's do one final question here.

We know the area of the triangle.

Let's use the formula, area multiplied by two, 40 multiplied by two equals 80.

That's the area of the parallelogram, the rectangle in this case.

And then 80 divided by 10 equals eight.

So that missing perpendicular height would be eight centimetres.

Very well done on your amazing achievements and accomplishments today.

You've been absolutely brilliant.

Why don't you give yourself a pat on the back? I hope you have a great day, whatever you've got in store, hope we get the chance to do some more maths together in the near future.

But until then, take care and goodbye.