Lesson video

Hi, welcome to today's lesson on using the formula for the area of a triangle.

By the end of today's lesson, you'll be able to use the formula for the area of a triangle to calculate missing side lengths.

Our lesson has two parts to it today, and the first part is calculating the area of a triangle.

This will be getting used to using that formula that we derived in our previous lesson.

To calculate the area of a triangle, the base and the height are used.

It's very important that the base and the height of our triangle are perpendicular to each other.

We worked out why this was important in our previous lesson on the area of a triangle.

Here are two examples showing what we mean by the base and the perpendicular height.

The first example on the left shows the base and in the perpendicular height is marked outside of the triangle because the opposite vertex overhangs the base so it's clearer if we mark the perpendicular height outside of the triangle.

In the second triangle on the right, we can see the base and the perpendicular height marked there as well.

And again, that right angle has been used to show that they're perpendicular to each other.

Now in the previous lesson we talked about defining the base because it was one of our key words.

And I said to you that some people might think the base means the bottom side.

Well, you can see in this example the base is not the bottom side at all.

It's not the side that triangle is sitting on.

Remember the base and the height must be perpendicular to each other, so any side of the triangle could be called the base.

We know which one it is when we can see a perpendicular height marked on.

Let's do a quick check.

Here is a triangle and you can see that there are four lengths marked on it.

What I want to know is the base and the height of this triangle are what? So in other words, which pair of lengths represent the base and perpendicular height? Pause the video now while you make your selection.

Welcome back.

Which one did you pick? If you picked B, then you chose correctly.

We know that the perpendicular height and the base are perpendicular to each other, so we are looking for two lengths that meet it a right angle.

Now if you thought 16 millimetres was the base, we have a quick check and there's no length there that's perpendicular to it, so 16 can't have been the base.

Well that means that we are down to just three measurements left.

If 15 millimetres is the base, we look to see if there's a length marked perpendicular to it.

And there is, it's the 12 millimetre length.

So the 12 and the 15 must be my base and perpendicular height pairing.

We know it's not the 10 millimetre length because although it is perpendicular to one of the sides of the triangle, we dunno the length of that side so it can't be part of our pairing.

Now we're going to go through calculating the area of this triangle.

We've just checked that we can definitely identify the base and the perpendicular height, which we'll need to do if we're going to calculate the area.

Here we have a triangle with four lengths marked on.

We need to identify which ones we actually want.

Remember, the area of our triangle is equal to the base times the height divided by two.

Which of these sides therefore do I need? That's right.

I need the six metre side and the five metre length that's marked because these two are marked as perpendicular to each other.

And we can see that that perpendicular distance of five metres runs from the base to the opposite vertex.

So we are multiplying six by five and then remember, because the area of any triangle is half the area of a parallelogram, we must divide by two.

Six multiplied by five and divided by two gives us a total of 15.

It's now your turn.

Calculate the area of the triangle you can see on the right.

Pause the video while you do this.

Welcome back.

Let's see how you got on.

The area of the triangle is equal to the base times the height divided by two.

Did you correctly identify what the base and the perpendicular height were? You should have put six multiplied by three and then divided by two.

Well, six times three is 18 divided by two gives us, that's right nine.

Well done If you've got nine, if you didn't reach that, can you see where you went wrong? Did you check that the two lengths that you wanted to multiply together were perpendicular to each other? And that that perpendicular height went from the base to the opposite vertex? It's now time for your first task.

What I'd like you to do is to calculate the area of each triangle.

Remember, the area of the triangle is found by multiplying the base by the perpendicular height and then dividing by two.

Pause the video while you complete this task.

Welcome back.

Let's now look at the next part of the task.

For this part of the task, I'm asking you to write an expression for the area of each triangle.

Now, if you've completed our algebra unit, you know that you can write an expression for the area because we don't have numbers here we have a letter to represent a generalised number.

So what I'd like you to do is use the algebra to write an expression for the area.

In other words, what would I multiply together in order to calculate the area? Don't forget that dividing by two.

What would an expression for the area look like for each of these triangles? Pause the video now while you have a go at writing an expression for the area of each triangle.

Welcome back.

Let's go through this and see how you got on.

First I asked you to calculate the area of each triangle.

For triangle A, you needed to identify the base and the perpendicular height.

So what you should have done is multiplied 2.

5 by four and either multiplied by a half or divided by two.

Remember, multiplying by a half is the same as dividing by two.

2.

5 times four divided by two gives us an answer of five.

Let's look at B.

In B, the base is five and the perpendicular height is two.

So to calculate the area we have two times five divide by two, giving us again another answer of five.

In C, the base and the perpendicular height are three and four and it doesn't matter which way round.

Remember either of these could be the base and either would be the perpendicular height.

The important bit is that five is not a length we want to be using here, so three times four and halved gives us six.

Now in D, we can see two perpendicular lengths that are marked on, but which is the one we actually want? That's right, we want the 3.

5, because that the perpendicular distance from the base, which is 4.

2 to the opposite vertex.

So we should have calculated 3.

5 times 4.

2 and then halved, giving us an area of 7.

35.

Well done if you didn't get caught out there.

Now let's look at E and F.

Remember in this one you had to write me an expression for the area of each triangle.

In other words, which lengths did you want to use.

For E, if I want to calculate the area of that triangle, I'll need to multiply the length that our length B and length D, so it'll be B times D, and then I would divide by two and there's my expression.

You could also have put half BD.

That would've been fine too.

'Cause remember, multiplying by half is the same as dividing by two.

Now let's look at F, and here is where we have slightly more complicated terms in that there are coefficients and there's more than one letter there.

Not to worry though it works the same way.

So let's identify our base and our perpendicular height.

Well, our base will be 5DA.

We know that because we can see that there's a distance marked three that goes from the base to the opposite vertex and it's at right angles.

Therefore 5DA times three, well three times five is 15, so 15, lots of DA all divided by two.

Well done if you've got that right.

If you've written three multiplied by 5DA divided by two, that's also right.

Remember, there are lots of equivalent ways to write the same thing.

It's one of the benefits of maths.

Lots of different ways to say exactly the same thing.

It's now time for the second part of our lesson.

And in this section we're going to be calculating a side length of a triangle given that we know its area.

So still using that formula but now to calculate something different and we're gonna see how that works.

We're gonna start off by calculating the height of the triangle that you can see on the left.

We are told that the area of this triangle is 33.

25 and we can see that there's a base of seven.

Well, let's start with what we do know.

We know that to calculate the area of a triangle, we do the base times the height and then divide by two.

So seven times something, whatever our height is, divided by two will give us 33.

25.

Well, let's work out what I can do.

I can work out what's seven divided by two or seven multiplied by a half is, that's 3.

5.

So now I've got 3.

5 times something equals 33.

25.

Now you can either do this with division.

Or you could use your calculator and type in 33.

25 divided by 3.

5, to find our perpendicular height, which is 9.

5 metres.

So as you can see, same formula just working out something different rather than the area, but we can still use it.

Now it's your 10.

I'd like you to calculate the base of the triangle you can see.

You're told that the area of your triangle is 24 and it has a perpendicular height of eight.

Now the perpendicular height wasn't marked on there, but I'm letting you know it is indeed perpendicular so you can use it, don't worry.

Pause the video now while you calculate the length of the base.

Welcome back.

Let's see how you got on.

Remember, the area is found by multiplying the base by the height and either multiplying by half or dividing by two.

Well, we know that the area is 24, so 24 is equal to the base times by eight times by a half.

Oh I can work out what half of eight is, that's just four.

So 24 is equal to four, multiply whatever the base is.

What do you multiply four by to get to 24? That's right, six.

So the base must be six metres in length.

Well done if you got that right.

If you weren't to certain about your four times table, then obviously you could have used a calculator to divide as well.

It's now time for the final task.

For each of these triangles, I'd like you to calculate the missing side length.

Remember what the area of a triangle is, it's half of the base times by the perpendicular height.

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

Welcome back.

It's now time for part two.

Here you have two questions, D and E.

In each of them I'm asking you something to do with missing side length.

D gives you the area of the triangle and tells you what its height is.

And what I'd like to know is what's the length of its base.

In E you have another triangle that has the same area, but in this one I'd like you to tell me what you think the length of the base and the height could be.

Try to come up with as many pair of lengths as you can, maybe some that no one else has thought of.

See what you can do.

Pause the video now while you complete D & E.

Welcome back.

Let's go through these.

In A remember, we were told that the area was eight and the perpendicular height is two.

So we know that eight equals a half times two times whatever the length of A is.

Well half of two is just one, so A must be eight.

In B, we're told the area is 10.

125.

And we know that one of the lengths is 4.

5.

Well, let's put that information into our formula.

10.

125 equals half of 4.

5 multiplied by B or half of 4.

5 is 2.

25.

I can now use my calculator to work out the length of side B, which is 4.

5.

In part C, we're told the area is 10.

5.

Now we have three lengths that are marked here.

We need to identify which one is perpendicular to C.

That's right, so the length of seven.

So we know that 10.

5 is equal to half of seven multiplied by length C or half seven is 3.

5.

10.

5 divided by 3.

5 is three, so C is three.

Now D, in this one we didn't have a triangle drawn for us, but you could have drawn it if it would've helped you.

Now we're told the triangle has an area of 120 and a height of 12.

We want to find the base.

So we use our formula, 120 is equal to half of 12 multiplied by D or half of 12 is six.

So 120 equals six lots of C.

120 divided by six is 20, so C is 20.

Now in E, I said that we had a triangle that was different to our first one, has an area of 120 as well.

What could the lengths of the base and the height be? Try to come up with as many pair lengths as you can.

Are yours going to be unique in that no one else thought of them? What you might have realised, of course, is that the area of a triangle is half of the base times the height.

So if I double the area and make it 240, then actually I'm looking for any pair of numbers that multiply to make 240 and they would be base and my height.

So for example, you might have said 24 and 10 that would've worked, so would've 120 and two.

So as long as your numbers multiply together to make 240, then you'll be right.

Let's summarise what we've learned today.

The area of any triangle can be found by multiplying the base of the triangle by its perpendicular height.

And the formula for finding the area of a triangle can also be used to find a missing side length in that triangle.

Of course, that missing side length has to be either the base or the perpendicular height.

Well done.

You've worked really well today.

I look forward to seeing you in our next lesson.