Lesson video

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

By the end of today's lesson, you'll be able to derive the formula for the area of a triangle.

Now, in our lesson today, we're going to be using some specific mathematical terminology.

For two words that we're going to be using today, in particular, our base and the phrase perpendicular height.

Now, you may have heard these words before, and it's excellent if you can recall their definitions, but just in case, let's recap them.

What I'd like you to do is to take a moment and write down what you think the definitions for each of these words are.

Pause the video now while you do this.

Welcome back.

Let's see if your definitions match mine.

The base is the side which is perpendicular to the height.

Now, that may have confused you.

You might have written down that the base was the bottom of the triangle.

That's not technically true.

Depending on the orientation of your triangle, the side that we're going to refer to as the base may not sit at the very bottom of the triangle.

It's worth therefore, keeping in mind that base does not necessarily mean bottom.

For a triangle, it is the side which is perpendicular to the height and you'll see lots of examples of this both in this lesson and in our future lessons within this unit and in fact across all the units we're gonna be studying.

The perpendicular height is the perpendicular distance from the base to the opposite vertex.

In other words, once we know which side of our triangle is our base if we draw a line at 90 degrees to this base to the opposite vertex i.

e the corner that isn't part of our base, that distance is known as the perpendicular height, and it's going to be very important for what we're doing today.

Now we've got those keywords.

We're ready to start our learning.

Let's start with our first section.

In our first section we're going to be reviewing area and the perimeter of a triangle.

Now, in primary you will have worked with area and perimeter of triangles before, so it's important that we recap this just before we move on to actually deriving the area of a triangle.

Let's review.

The area of a triangle can be found by multiplying the base and the perpendicular height of the triangle and then we divide by two.

Here we have our triangle.

We can see that it has two measurements listed on it, a four unit length and a three unit length.

The area of this triangle is found by multiplying these two lengths together and then dividing by two.

Remember how we said the base and the perpendicular height were what we needed? Well, which of those two lengths, the four and the three, which of them do you think is the base? Does it have to be a particular length? Well done if you said no.

The base could actually be either of these two lengths.

As soon as we say which one the base is, the other length becomes our perpendicular height.

We can see that these two lengths meet at right angles.

In fact, the right angle is actually marked on our triangle for us.

That little square there that you can see at the bottom right hand side of the triangle is in fact showing us that those two lines are perpendicular and that's how we know they're the two lengths to multiply together.

Don't forget, of course it's a triangle, so we divide by two.

Now you might be wondering, why do we divide by two? Hold on.

That's in the second section of our lesson today so we will see very soon.

Now, it's sometimes easy to show the perpendicular height outside of the triangle.

Here's an example where we have our base marked on, and in this case the base is at the bottom and the perpendicular height is listed as five.

We can see it's perpendicular to where the base is.

Again, because we've marked that right angle on.

We've drawn it outside of the triangle 'cause well, we want to show it's straight, remember from the base to the opposite vertex.

We wouldn't be able to show that if we'd drawn a line inside the triangle so outside makes it nice and clear.

To find the area of this triangle, therefore, we multiply the base by the perpendicular height so seven times five, and then we divide by two, giving us an area for this triangle as 17.

5.

Let's just have a quick check-in.

Do these two triangles that you see on the screen have the same area? Once you've worked that out, I'd like you to explain your reasoning, so if you said yes, tell me why or tell me how you know.

If you said no, explain why they don't.

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

Welcome back.

Did you say that they had the same area? You're perfectly right.

They do.

They have the same base and the same perpendicular heights so they must have the same area.

It does look a little odd though, doesn't it? They don't look like they do, but they must.

Does this animation help? It starts off by showing our original triangle and then all we're doing is dragging the top vertex.

Notice how our height isn't changing as we do this.

Showing that our two triangles have the same area.

It's now time for your first task.

Here on the screen you can see four triangles.

For each triangle, I want you to tell me if you have enough information to calculate the area, the perimeter, both area and perimeter, or neither.

Please make sure you explain how you know.

Pause the video now while you write down your answers to this first question.

Welcome back.

Now that you've written that down, you are ready for this second part.

For the triangles where you said something wasn't possible so either you couldn't work out both area and perimeter or you couldn't work out just one.

Please write down the minimum additional information that you need in order to calculate the area and the perimeter.

Pause the video now while you do this.

Let's go through our solutions.

In triangle A, you can calculate the area.

We have a base of 2.

5 and a perpendicular height of 4 but you can't work out the perimeter.

One of the side lengths is missing and remember, perimeter is the distance around a 2D shape.

Without that side length, we can't work out perimeter here.

Let's look at B.

In triangle B, you can calculate the perimeter because you know all the side lengths.

However, you cannot calculate the area.

For the area you would've needed the perpendicular distance from whichever side was going to be the base to the opposite vertex and that isn't marked here.

You might have thought that the four centimetre line and the two centimetre line were perpendicular to each other.

You're right, it does look like they might be, but remember we can't assume this.

We haven't been told it and it's not marked on the diagram.

That means we can't work out the area.

Let's look at C.

In C, you can calculate both area and perimeter.

We know the three side lengths and we can see that two of them are perpendicular to each other, which means we can do the area.

What about in D? In D, you can't work out area or perimeter.

We can't do the perimeter because we have two missing side lengths and we can't do the area because although we can see there's a perpendicular distance marked on there, we don't have the side that would be the base, so no area for us there either.

It's now time to move on to section two.

Remember how I said that we'd been working with the formula for the area of a triangle but we didn't actually know where that'd come from.

Now it's time to actually derive it ourselves.

In other words, why does it work like this? Why do we find the area of a triangle by doing the base times the perpendicular height and then dividing by two? Let's explore and find out.

We can derive the formula for the area of a triangle through an investigation.

That's right.

You might have thought investigations were only for science lessons.

I think not.

They're just as important in maths as well, so let's get ready and do this investigation.

You're going to need a piece of paper.

Now, you'll want it to be a reasonable size because you need to fold it in half.

A4 would be perfect.

If you've got something a little smaller, that's fine too.

Just make sure it's big enough for you to be able to fold in half.

Once you've folded it in half, you're ready for the next step.

In this step I'd like you to draw a triangle on your piece of folded paper.

Now you can see I've done an example on the screen.

Your triangle doesn't have to look like mine at all.

Your triangle can look very different.

Just make sure it's a triangle so three straight sides all joined.

Got it? Then you're ready for the next instruction.

You now need to cut out your triangle.

Because your paper is folded, when you cut it out, you'll actually be cutting both sides of paper, so the one that your triangle's drawn on and the one underneath.

Therefore, when you've cut out your triangle, you'll actually have two triangles, the one from the top half of the paper and the one from the bottom half.

Your triangles should be identical as long as you've cut carefully.

Once you've done this, you'll be ready for the next instruction.

I'd like you to put your two triangles next to each other.

You can flip them, rotate them, do it to just one, do it to both.

It's entirely up to you.

What I'd like you to do though is write down which of our quadrilaterals you make when you put these two triangles next to each other.

You may be able to make more than one, so make sure you write down your list.

Don't miss any out.

Pause the video now while you do this.

Welcome back.

Let's see which quadrilaterals you made.

Now, depending on the triangle that you drew, the following quadrilaterals are all possible, so if you do triangles like I did, then you could have made a kite, a parallelogram, you could also have made an arrowhead.

If you'd drawn a triangle that had a right angle in it you may well have been able to make a rectangle.

If your triangle had lengths that were the same, then you might have been able to make a rhombus or even a square.

It all depended on the triangle you drew.

If you didn't make some of these quadrilaterals feel free to try with a different triangle and see if you can make them.

Now it's time for the final part of our investigation.

I'd like you to take your two triangles and try to make each of the six quadrilaterals that you can see below.

Which ones can you make? Then try some other triangles.

Which of the quadrilaterals can you make? Can you see where this is going? Try experimenting with lots of different types of triangles.

Which of the quadrilaterals that you can see below can always be made? Pause the video while you investigate.

Welcome back.

So which quadrilateral could you always make? Regardless of the triangle drawn, you can always make a parallelogram.

Rectangles and rhombuses are special types of parallelograms and don't forget that squares are special types of rhombuses so actually you've been making a parallelogram with every single triangle.

Hmm, that's interesting.

I'm sure we know some facts about parallelograms that might help us here.

The area of a triangle is half the area of a parallelogram.

You've just experimented and seen that two identical triangles can always be placed together to create a parallelogram, so let's derive our formula formally now.

The parallelogram on the screen has a length B and another length H that are marked.

B is representing the base of our parallelogram, and H represents the perpendicular height.

We know how to find the area of a parallelogram.

In fact, we reviewed it a couple of lessons ago.

The area of a parallelogram is equal to the length of the base multiplied by the length of the perpendicular height.

In other words, the base times the perpendicular height.

Now, we can write this shorthand using algebra.

In this case I would write BH.

Remember that stands for B multiplied by H or B times H, and it shows that we are multiplying these two lengths together.

Here is a triangle.

Two of these triangles were put together to form the parallelogram and that's why that dotted line's there on the parallelogram to let you see where the two triangles meet.

The base of our triangle is the same as the base of our parallelogram and H is still the same perpendicular height.

The area of our triangle is therefore equal to the base times the perpendicular height but it's half of our parallelogram so I'll need to either multiply by a half or divide by two.

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

Now that's our formula written out using words but remember we're mathematicians.

Let's try to generalise and use our algebra.

This is one of the ways the formula can be written.

We have BH for B times H.

Then we're dividing by two.

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

And that's it.

We've just derived the formula for the area of a triangle.

We're not just using a formula because our teacher told us.

We've actually seen how that formula can be worked out, and you did that.

You just investigated and discovered a formula just by doing some cutting and rearranging some shapes.

Maths isn't always pencil and paper and lots of writing.

Maths is creative and that's what you did today.

You took two triangles and created various quadrilaterals and then you spotted a pattern and you looked how to generalise.

Let's just check before we end today's lesson.

True or false, the area of a triangle is always half the area of a rhombus.

Is this true or false? Don't forget, you'll need to justify your answer so check the reasoning for A and the reasoning for B and see which one supports your decision as to whether the statement is true or false.

Pause the video now while you make your selection.

Welcome back.

Let's see how you got on.

It's false.

We know that two identical triangles placed together make a parallelogram, not necessarily a rhombus.

The area of a triangle is always half the area of the parallelogram that's formed by those two identical triangles being placed together.

Let's summarise our learning for today's lesson.

The area of any triangle is half the area of the parallelogram made from two copies of that same triangle, and we saw that in our investigation.

Well done.

You've worked really hard today.

I hope you enjoyed the investigation and I look forward to seeing you in the next lesson.