Lesson video

Hi there, my name is Mr. Tilstone.

It's great to see you today.

Hope you're having a good day, and let's make it even better by having a great math lesson.

So if you're ready, I'm ready.

Let's begin.

The outcome of today's lesson is I can explain how to calculate the area of a triangle.

You might have had some very recent experience of calculating the area of a parallelogram.

I wonder if the two are linked.

Let's find out.

Our key words today, we've got three.

My turn, parallelogram.

Your turn.

My turn, base.

Your turn.

And finally my turn, perpendicular.

Your turn.

I'm sure you are familiar with those words, but it's worth a little recap, a little reminder.

They're very important words and they're going to come up a lot today.

So let's have a look.

A parallelogram is a quadrilateral with two pairs of parallel and equal sides.

You can see an example there.

The base is a side which is perpendicular to the shapes height, and you can see that on the parallelogram.

And two lines are perpendicular if they meet at a right angle.

And you've got a couple of examples there on your screen.

Our lesson is split into two cycles today.

The first will be the perpendicular height of a triangle, and the second calculate the area of a triangle using a formula.

So if you're ready, let's start by thinking about the perpendicular height of a triangle.

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

Have you met Alex before? He's here today to give us a helping hand with the math.

Some very good ears too.

"Just like parallelograms, "all triangles have a perpendicular height, "a line that can be drawn from a base "to the height of the opposite vertex." So can you remember any recent experience you've had of finding the perpendicular height in parallelograms? Triangles have them too.

And it can be shown, just like in parallelograms, in various positions.

So let's have a look.

So we've established a base on this triangle, and that would be the perpendicular height.

So can you see it's formed a right angle.

So it's very similar to the perpendicular height of a parallelogram.

Here's the same triangle with the same base, but the perpendicular height's been marked somewhere different.

It's still a perpendicular height though.

It's exactly the same.

And here's another one.

What do you notice about that one? That one's outside of the triangle.

It's still the perpendicular height.

The base is a side from which the perpendicular height is found.

So although you might just think of the base as being the sort of bottom of a shape, it can be the opposite as well.

So we've got the base here, there's a perpendicular height.

The perpendicular height measurement will change depending on which side is chosen as the base.

The base is the side from which the perpendicular height is found.

So you can see we've got a different base here.

So it's not necessarily the longest side or the shortest side or anything like that.

It can be any side of a triangle.

It's useful to rotate the triangle so that the base is horizontal or vertical to ensure that the measurements are accurate.

So if on your page you can see a triangle and it's not got horizontal base, you might want to turn it around, so it has, but you don't have to do that.

That's just a tip.

So there's a perpendicular height of this particular triangle.

You can see that right angle that's been formed.

The base is a side from which the perpendicular height is found.

It's useful to rotate the triangle so that the base is horizontal or vertical to ensure the measurements are accurate.

So again, like here we've got the base, and here we've got a perpendicular height.

The triangle does not have to be rotated though for the perpendicular height to be found from a base.

So in this case, we're going to leave it so that the base is not horizontal.

And here we go.

You might have to tilt your head a little bit to see this, but you can see we've established our base.

And from that base we can then draw a perpendicular line to form the perpendicular height.

"When a triangle is right angled, "two sides can be used to show the base "and perpendicular height." So here's a right angled triangle, you might recognise that.

There's a base.

And this is a perpendicular height.

So can you see, just like before it's forming a right-angle from the base.

"In an isosceles triangle, "using the unequal side as the base "can often be the most efficient way "to find the perpendicular height." So this is an iso lease triangle.

What can you remember about those? Two side lengths the same, one different.

"Rotating the triangle so that the base "is horizontal or vertical, makes it more efficient." So it's a really top tip.

So in this case, look, we've shown the base, but it's actually vertical.

It's still the base.

And from that base we can establish that perpendicular height.

Let's see how much of that has sunk in.

Is Alex correct? He says, "I have drawn a line "to show the perpendicular height of this triangle." Has he though? What do you think? Can you explain, pause the video.

Well, I think good for you Alex, because you've had a good go at this and I think you've found some good things there.

He's established a base, for example, that's not showing the perpendicular height though, is it? Would you agree? No.

The line must form a right angle from the base to the height of the opposite vertex.

And it's not doing that.

That's a side length, but it's not the perpendicular height.

So let's try and give it a perpendicular height.

Here we go.

Now we've got a line that forms a right angle, that's a perpendicular height that goes to the opposite vertex.

Time for some independent practise.

Let's see what you can do with these questions.

So number one, "Tick the triangles, which correctly show "the perpendicular height of the triangle "to the given base." So I can see looking at that, some are correct, some are showing a perpendicular height and some aren't.

They're in different kinds of orientations.

And again, remember that top tip, you might want to rotate your page if you need to.

So number two, those that are incorrect, can you correct them? Can you make it so that they have got a perpendicular height? Number three, "Determine a base for each triangle "and then draw a line to show the perpendicular height." There's going to be more than one answer to each of these.

Remember, it may be easier to rotate so that your base is horizontal or vertical.

Good luck with that.

Off you go and I'll see you soon, and I'll give you some answers and some feedback.

Welcome back.

How did you get on, finding those perpendicular heights? Do you think you've got that? Is that nailed in? Let's have a look.

So the triangles, which correctly show the perpendicular height are as follows.

This one does.

We've got a base, we've got a perpendicular line that forms a perpendicular height to the opposite vertex.

That one was pretty straightforward, the base was already horizontal.

And we've got this one this time the base is vertical, but it's still a base and we've still formed a perpendicular height from that, forming a right angle.

You might to even drawn the little right angle squares in to show that.

Number two, "Correct those that are incorrect." So for the first one, we could redefine that base.

So change that base to a different position and that would make that correct.

'Cause the line was there already to form a perpendicular height but not the base.

So that's one way to do it.

There's other ways too.

And for this one, making the height line perpendicular to the base would make that one correct.

So you might have done something like that or against something different.

And number three, "Remember the perpendicular height "is dependent on the base that you chose." So you might not have these exact answers.

These are just some examples.

And p.

h.

stands for perpendicular height here.

So we've got a base, we could establish that horizontal line is the base.

And then from that we can form a perpendicular height to the opposite vertex.

And again, the same with B.

That's the right angle triangle, so maybe we could have the sort of bottom of the triangle as the base.

And then the perpendicular height again goes to the opposite vertex.

And then for C, well C's not horizontal or vertical.

You might have made it that way by rotating your page or you might not have done either.

But we've established a base and from that we can establish a perpendicular height to the opposite vertex.

So that's just a few possible examples.

"Did you notice that A is isosceles? "And so the unequal side length is best to use "to show the height." That was a top tip from earlier.

So well done if you spotted that.

"Did you notice that B is right angled.

"And so two sidelines can be used as a base "and perpendicular height." And remember that's not always the case, is it? Okay, are you ready for the next cycle? That's calculating the area of a triangle using a formula.

"What do you notice about the rectangle?" And this might be jogging your memory about some recent learning that you might have done.

What do you notice? Have a look at the rectangle.

How has it been composed? How could it be decomposed? "A rectangle can be composed of two congruent, "right-angled triangles." So they're the same as each other, those triangles, they're just in different rotations.

So that's two congruent right angle triangles together, making a rectangle.

Does that ring a bell? Have you explored that before? Here we go.

So we're decomposing that.

And again, you can still see very clearly those two congruent right angled triangles.

Hmm, I wonder if this is making you think how you could calculate the area of a triangle.

Because you may hopefully already know how to calculate the area of a rectangle, hmm.

The area of a rectangle can be calculated by multiplying the base by the perpendicular height.

Okay, you remember that? Hopefully you've done that quite a lot.

So base times by perpendicular height gives you the area of a rectangle or indeed any parallelogram.

But what could be said about the area of one of those triangles? Hmm, you notice, remember they're congruent, they're the same.

What's happened here? What do you notice? It's half the area of the rectangle.

So how could we use that to work out the area of a triangle? A right angle triangle specifically.

Area equals base times by perpendicular height divided by two.

So we made that rectangle base times perpendicular height and then we halved it, we divided it by two.

And that's a formula.

Right, let's say that together.

This is a formula for calculating the area of a triangle.

Say it with me.

Area equals base times by perpendicular height divided by two.

Brilliant.

Just you say it now, off you go.

Let's have a check for understanding.

Let's see what you've managed to understand there.

So use the formula, and that's there at the bottom of the screen, area equals base times by perpendicular height divided by two.

Use that formula to calculate the area of one of those triangles.

So we've got a base established that's five centimetres and we've got a perpendicular height established, that's four centimetres.

What's the area of one triangle? The arithmetic there is fairly gentle, so let's see how you get on.

Pause the video.

How did you get on? Did you multiply five by four to get the area of the rectangle.

That gave you an area of 20 centimetres squared.

Did you then divide that 20 by two to get the area of one triangle? That's 10 centimetres squared.

Just like this.

Well done, if you've got 10 centimetre squared, you're on track.

This can be used for any right-angled triangle.

So that's a rectangled triangle.

No matter what they look like as long as they're a right angle triangle, that same formula works, area equals base times by perpendicular height divided by two.

There we go.

So that's the imaginary other triangle that will go with it.

The imaginary congruent triangle that would make it into a rectangle.

We can't see it, we can see it there, but you can imagine it.

When you see a rectangle triangle, try to imagine or visualise a congruent right angle triangle forming a rectangle.

And when one of those triangles is removed, the base is still known and the perpendicular height is still known.

So we don't have to see it as a rectangle, we can just see it as a triangle.

And we still know the base and we still know the perpendicular height.

So we can still use that formula.

Let's have another quick check, shall we? "Calculate the area of this triangle?" The formula is still there.

You don't need to remember it, although perhaps you already have.

Perhaps you've learned it off by heart already.

But there is a triangle.

We've got the base established, we've got the perpendicular height established.

So what is the area of that triangle? Pause The video.

The arithmetic in that question was fairly gentle, wasn't it? So we had to do eight multiplied by 10.

That's the timestables fact.

That gives us 80 centimetre squared.

That's the area of what would be the rectangle.

And then we're halving that to give us the area of that triangle.

We dividing it by two.

And that's again, fairly gentle arithmetic, isn't it? Hope we can do that.

80 divided by two or half is 40.

So the area of the triangle is 40 centimetres squared.

And if you've got that, you are bang on track, you're doing really well.

"What about triangles that do not have a right angle?" Do you think that formula would still work? So here we've got a scalene triangle.

All three of the sides are different lengths.

Hmm.

Can they be used to compose a rectangle? Hmm, if you had another congruent triangle exactly the same as that one, could you make a rectangle from it? No, but you can make a parallelogram.

A parallelogram can be composed of two congruent triangles.

And there's more than one way to do that.

Here there are two congruent scalene triangles and together they make a parallelogram.

"One triangle is half the area of the parallelogram." And can you remember how to calculate the area of a parallelogram? "A parallelograms area can be calculated "by multiplying its base by its perpendicular height." Just like with a rectangle, it's the same formula.

and in one triangle is half the area of the parallelogram.

So it is exactly the same as it was before when we were investigating the right angle triangles.

The same formula applies.

We can multiply base by perpendicular height and divide by two.

And here is that formula.

So again, I'll say it, we'll say it, you say it.

Area equals base multiplied by perpendicular height divided by two.

Let's say it together, area equals base multiplied by perpendicular height divided by two.

You, off you go.

Fabulous.

So just the same as before.

This formula can be used for the area of any triangle.

Because any triangle, when you combine it with a congruent triangle, makes a parallelogram.

Any triangle could be half of a parallelogram.

The base is still known.

The perpendicular height is still known.

We don't need to see the whole parallelogram to know that.

You can visualise it if you need to.

Let's have another check.

"What is the area of this triangle?" So we've got our base established.

We've got our perpendicular height established.

Can you work out the area of the triangle? Pause the video.

How did you get on? Did you spot that? We had a times tables fact there, 12 multiplied by seven.

Hopefully that was automatic for you, if you know your tables and you knew straight away that was 84.

So here's our formula area equals base multiplied by perpendicular heights.

That's 84 centimetres squared and then divided by two, halved.

So we need to do half of 84, which is hopefully quite straightforward.

84 divided by two equals 42.

So the area of that triangle is 42 centimetres squared.

Congratulations.

If you've got that, you're gonna be ready for some independent work.

And here is that learning.

So number one, "What is the area of each of these triangles?" Some of them have got some clues on there, some of them have got some red herrings on, some extra information that you don't need.

So look out for those.

So be careful if there's a side length, for example, that you don't need.

And we've got some more here.

You might notice something a bit different about it.

Look very carefully, very closely.

Hmm.

Number two, "Measure the base and perpendicular height "of this triangle." What's the area of the triangle? So remember all those rules about measuring side lengths.

Can you work out the area of that triangle? And number three, "Is Alex correct? "Explain." He says, "I can calculate the area of a triangle "if I have two measurements." Hmm, is that right? What do you think? See if you can explain that one.

Good luck with all of that.

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

Welcome back.

How did you get on? Number one, "What is the area of each of these triangles?" Well, for that first one, we can see that we've got our base of nine centimetres, our perpendicular height of 10 centimetres.

That's a nice easy one to multiply together.

That's 90 centimetres squared.

So that's the area of the rectangle.

But we need to then halve that.

So 90 centimetres square divided by two equals 45 centimetres square.

And for B, did you notice there were three measurements there? You only needed two of them.

The side length, we don't need that.

12 centimetres is not going to help when establishing the area of the triangle, we need the base and the perpendicular height.

So that gives us 11 multiplied by 10.

Nice and straightforward, hopefully, that's 110 centimetres squared, that's the area of the parallelogram, which we then half, we divide it by two and that gives us 55 centimetres squared.

And then for C, the base is not horizontal there and it's not vertical either.

You might have tilted your page so that it was or you might not have done.

But either way we've got our two measurements, our base 12 centimetres, our perpendicular height is 15 centimetres.

Multiply those two together.

That's not a times tables fact, but that's 180 centimetres squared.

You might have done 10 times 15 and two times 15 and combine those together.

But 180 centimetres squared, half of that is 90 centimetres squared.

And that's the area of the triangle.

And for D, this time it was in metres, did you spot that? But the same rules apply.

Now we've got our base that's 20 metres.

Did you notice something different about the perpendicular height? It was outside the triangle, which it can be.

So we extended the base to show that.

So 20 metres multiplied by 25 metres equals 500 metres squared.

That's the area of the parallelogram.

I did 10 metres times 25 and doubled it.

And then 500 metres squared halved divided by two equals 250 metres squared.

And that is the area of that triangle, that scalene triangle.

And for E, did you notice that the units were different? One was in centimetres, one was in millimetres.

Well, we can convert, can't we? So instead of thinking of it as 80 millimetres, why don't we think of it as eight centimetres? And then we've got eight centimetres multiplied by 11 centimetres.

Nice, easy one, hopefully, it's times tables.

That's 88 centimetres squared.

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

And then half of that is 44 centimetres squared.

So well done if you weren't fooled by the mixed units there.

And number two, measuring the base and perpendicular height of this triangle or the base is 13 centimetres.

Well done if you measured that accurately.

And the perpendicular height is four centimetres.

So we're gonna multiply those two together to give us 52 centimetres squared.

That's the area of the parallelogram.

And then we divide that by two or we halve it.

And that is 26 centimetres squared, is the area of that scalene triangle.

And then was Alex correct? What do you think? I think he was half correct, don't you? The area of a triangle can be calculated when the base and perpendicular height are known.

So that is two measurements.

If for example, the base and the other side length, which is not the perpendicular height known, that's not enough information to calculate the area.

So sometimes if you are using the side length as one of the measurements, that's not correct.

So here's an example.

Here's two measurements from the triangle, two different side lengths.

We've got a base, but we haven't got a perpendicular height.

So in that case, two was not enough.

So it does depend on which two.

One's gotta be the base, one's gotta be the perpendicular height.

We've come to the end of the lesson.

Today's lesson has been explaining how to calculate the area of a triangle.

Hopefully you're starting to get to grips with that now.

"Any triangle, any triangle at all can be thought of "as half of a parallelogram, "including right angle triangles, "which are each half of a rectangle." And remember, a rectangle is a parallelogram.

"A triangle has half the area of a parallelogram "with corresponding base and perpendicular height.

"The formula for calculating the area of a triangle "is therefore this." Are you ready for it? One more time.

Area equals base multiplied by perpendicular height, divided by two.

Well done on your accomplishments and your achievements today.

You've been absolutely fantastic.

I really hope I get the chance to spend another math lesson with you in the near future.

But until then, enjoy the rest of your day, whatever you've got in store.

Take care and goodbye.