Lesson video

Hello, my name is Dr.

Rolson and I am happy to be helping you with your learning during today's lesson.

Let's get started.

So welcome to today's lesson from the unit of perimeter, area and volume.

This lesson is called Checking Understanding of Area.

And by end of today's lesson, we'll be able to find the areas of triangles, quadrilaterals and composite rectilinear shapes.

Here are some previous keywords that you may be familiar with and we'll be revisiting again in today's lesson.

This lesson contains three learn cycles.

In the first learn cycle, we'll be finding the areas of rectangles, triangles, and parallelograms. And then the second, you'll be find the areas of trapezia.

And then in the third, we'll be finding the areas of composite shapes.

But let's start off with rectangles, triangles, and parallelograms. The rectangle below represents a field that is surrounded by a wall.

Which could be found by calculating the area of the field.

In this context, would it be the total amount of grass or the total length of the wall? Pause the video, have a think and press play when you're ready to continue.

In this context, the area of the field is the amount of space that is covered with grass, which in this case is 10,800 metres squared.

So area is the amount of 2D space that is taken up by a shape.

And finding the area of a shape always involves multiplying together two perpendicular length, somewhere within the calculation.

Now that multiplication of perpendicular length may be subtle within the formula of a particular shape, but it will be there somewhere.

Somehow or somewhere, there will be one length being multiplied by a perpendicular length, even if it's not always fully clear.

That's because area is measured in square units and the length of a square are perpendicular.

So to create the squares, we need to use perpendicular lengths.

And what we're thinking about when we find the areas and use square units is how many squares does this shape take up? If I was to take all the partially filled squares and join 'em all together, for example, how many squares of space would it take up? And the units we use really depend on what units of length you've got.

So if you are measuring length in kilometres, then our units for area would be square kilometres on.

If you're measuring length in metres, it'll be square metres.

And you may sometimes hear the units for areas said in different ways.

So someone may say square kilometres, or they may say kilometres squared, and it might say metres squared, or it might say square metres if it either way is absolutely fine.

So examples of units could be kilometres squared, metres squared, centimetres squared or millimetres squared, if working with the metric system, or it could say those as square kilometres, square metres, squared centimetres and square millimetres.

Or if working with imperial units you might hear miles squared or square miles or feet squared or square feet or inches squared or square inches.

There are different formula for calculating areas depending on what shape it is you're doing.

But all formulas include a multiplication of two perpendicular lengths.

For example, how would you find the area of each of these three shapes? We've got a rectangle, a parallelogram, and a triangle.

Perhaps pause the video at this point and just see if you can remember what are the formulas for each of those shapes and what numbers would you use to find the areas of each of these shapes? Pause the video, have a think and press play when you're ready to continue.

So the first shape we have a rectangle.

And the formula to find the area of a rectangle is width times length.

And this case we can see we've got a width of four and a length of five.

So we've got four times five, which is 20.

And because the units aren't specified, we can just put square units.

The second one, we have a parallelogram, and the formula for a parallelogram is base times height.

Now, sometimes people think that we're gonna do four times six in this situation because the four and six are both lengths on the sides of this parallelogram, but the four and six are not perpendicular to each other.

So we cannot make squares with lengths are not perpendicular to each other.

So we're gonna use the four and the five because they are perpendicular.

And the five is the height of the parallelogram.

So do 4 multiplied by 5 to get 20 square units.

And for the triangle, we can think of a triangle as being like half a parallelogram.

If we were to cut the parallelogram across, its diagonal.

So the formula for area of a triangle is half times base times height.

And once again, the base and height needs to be perpendicular to each other.

Now it doesn't matter necessarily which orientation they're in as long as they are perpendicular.

So, base and height in this situation are the five and the six.

If you struggle to see that, you can perhaps turn your head sideways so that the five is horizontal from your perspective, and you'll see that the six goes vertically in that case.

But the key thing is the five and the six are perpendicular to each other.

Half times 5 times 6 is 15.

And then we've got square units.

So let's check what we've learned so far, which is the formula to calculate the area of a triangle? Your options are A, B, and C.

Pause the video, have a go and press play when you're ready for the answer.

The answer is B.

The area is half times base times height for a triangle.

True or false, you can calculate the area of a parallelogram by multiplying together a and b? Pause the video, choose true or false and in one of the justifications, either A, you need to multiply together two lengths to find the area, or B, a calculation for area must include the multiplication of two perpendicular lengths.

Pause, have a go and press play when you're ready for the answer.

The answer is false because a calculation for area must include the multiplication of two perpendicular lengths.

Which two calculations could be used to calculate the area of the parallelogram? You got four calculations to choose from with options A, B, C, and D.

So pause the video, make a choice, and press play when you're ready for an answer.

The answers are, we could do A times C because those lengths are perpendicular to each other, or we could do B times D because those lengths are perpendicular to each other.

In each case we've got A and B, which represent sides on the parallelograms and C and D.

Well, those are perpendicular heights to those particular sides.

If you imagine, for example, the turning the parallelogram around that A was horizontal C would be going vertical upwards and would be the perpendicular height.

And we can see that more clearly with B times D.

So over to you now for task A, this task contains one question and here it is, you need to find the areas of each of these shapes.

Pause the video, have a go at all these and press play when you're ready to go through the answers.

Well done with that.

Let's now go through some answers.

So in part A we have a rectangle, which is six units by four units.

If we multiply those, we've got 24 square units, or you can even count the squares in that one and count that there are 24 squares.

In part B, we have a triangle with a base of five and it's perpendicular height of four and they're in centimetres.

We multiply those together and half them we get 10 centimetres squared.

In C, we have a parallelogram, we need to find those two perpendicular lengths, that's 9 centimetres and the 10 centimetres.

So we get 90 centimetres squared or square centimetres.

And in D, we need to find those perpendicular lengths.

So we have 8 times 8, which is 64 millimetres squared.

And in part E we have a parallelogram.

We need to find those perpendicular lengths.

Got 15 times 1, which makes 15 centimetres squared.

In part F we have a parallelogram and there are two different ways we can get our answer.

We can either do 12 multiplied by 4.

5 'cause they're perpendicular or six multiplied by nine 'cause they're perpendicular.

And either way we get the same answer of 54 metres squared.

Fantastic work so far.

Now let's move on to the second learn cycle, which is finding areas of trapezia.

Let's start off with though finding the areas of these three shapes.

We have a triangle, a rectangle, and another triangle.

The heights of all three shapes is six units.

Pause the video, find the area of each of them, and press play when you're ready to continue.

So the area of the third shape is six square units.

The area of the second shape is 24 square units and the area of the third shape is 12 square units.

Now these three shapes combine together to make a trapezium.

If you imagine just squeezing those shapes in from the outside until they are side by side, you'd have a shape that looks a bit like this.

Your triangle, your rectangle, and your triangle join together.

And we know it's a trapezium because it's a quadrilateral, a four sided shape, and it has exactly one pair of parallel sides.

That's a top side and the bottom side.

And we could compare the measurements between these shapes to get the measurements that go on a trapezium, for example, the height of all three shapes is six, which means the height of the trapezium must be six.

And then those bases on those three shapes, you've got two, four, and four they must add together to make the base of the trapezium, which is 10.

And the only measurement that contributes to the top of the trapezium is that rectangle.

So that must be four.

Also, based on this information, we can start to think now how would we find the total area of the trapezium? Pause the video and have a think about this and then press play when you've got an idea and are ready to continue.

A trapezium is made out of those three shapes, those three shapes combine together to make the trapezium.

Therefore, the area of the trapezium must be equal to the sum or the total of the areas of the three shapes.

So we can find the area of the trapezium by adding those together to get 42 square units.

Here we have two other shapes.

We've got a parallelogram and a triangle.

Now the height of each shape is six.

So pause the video, find the areas of these shapes and press play when you're ready to continue with them.

The areas of these shapes are 24 square units and 18 square units.

Now these shapes can also be combined together to make a trapezium.

If you imagine 'em pushing 'em together again until they are joined in that diagonal edge there, you'd have a shape that looks a bit like this.

You've got your parallelogram and your triangle.

Once again, we can put the measurements on there.

The height is 6 the base must be 10 because it's the 6 from the triangle plus the 4 which is at the bottom of the parallelogram as well as the top.

And then the top of the trapezium is just a four from the parallelogram.

Once again, how could we use the information we have here to find the area of the trapezium? Pause the video, have a think and press play when you're ready to continue.

If the trapezium it can be thought of as being the parallelogram and the triangle put together, then we can find the area of the trapezium by adding together the areas of the parallelogram and the triangle to get 42 units squared.

And here we have two triangles.

The areas have been given this time as 30 units squared and 12 units squared.

Once again, we can combine these to make a trapezium.

If we push 'em together, we've got now a four-sided shape with one pair of parallel sides.

The measurements, well, you'll have six for your height, that's the same as the triangles.

The base of the left triangle is the base of the trapezium and the top side of the right-hand triangle is the top side of the trapezium.

So those are our measurements and once again, we can find the area of the trapezium by adding together the areas of the two triangles because those together combine to make the trapezium.

So we get 42 square units.

So the areas of trapezia can be calculated by splitting it into other shapes.

For example, we could take a trapezium and split it into a triangle, rectangle and another triangle.

We could find the areas of all those three shapes separately, or we can combine those two triangles together to make a bigger triangle and overall triangle 'cause we'll have the same height.

We could split a trapezium into a parallelogram and a triangle like so here and find the areas of each of those add them together.

Or we can split a trapezium across one of its diagonals to create two triangles and find the areas of each of those.

Or we could split the trapezium into three triangles like so here and find the area of each of those triangles.

Or there is also a formula.

The area of trapezium is half times the sum of A and B multiplied by H.

Now those letters A and B, they represent the parallel sides of the trapezium and the H represents the height.

So we'll do the sum of the parallel sides, multiply by the heights and times by a half.

Okay, so now let's work through an example together and then shortly afterwards you'll have a chance to have a go at a similar one yourself.

Find the area of the trapezium by splitting it into smaller shapes.

Now I can split it in lots of different ways, but for this example, let's look at splitting it like this, a triangle, a rectangle, and a and another triangle.

Now when I look at these shapes separately, I've combined triangles A and C together.

And the reason for that is while I do know the measurements on the rectangle are four and five, what I don't know is what are the measurements on each individual triangle? I'm not sure quite what the base is on that triangle A and what the base is in triangle C.

But what I do know is if four from the base of the trapezium is taken up by the rectangle, there must be six left on that base of the trapezium for those triangles.

So altogether we know that the base of the triangle must be six units regardless of how much of that is for triangle A and C.

So that's why I've combined them together in this case.

To find a total area now, I can do four times five to get the area of the rectangle and I can do six times five and times a half to get the area of the triangle and then to get the area of the trapezium and add those together to get 35 square units.

So it's your turn now.

Find the area of the trapezium by splitting it into smaller shapes.

Pause the video, have a go at this and then press play when you're ready to go through it together.

Okay, let's see how we got on with that.

Once again, there are lots of ways you can split the trapezium up.

We could split it up as a parallelogram and a triangle.

We could split it into two triangles.

We could split it into a triangle, a rectangle and a triangle like we did on the example on the left.

Lots of different ways.

Let's go with this example.

If I've split into a triangle, a rectangle and a triangle, we can see the rectangle has measurements of three and four, the triangles, we don't know the individual triangles, but we know that we combine 'em together, the base must be seven and the height must be four.

Find the areas of each, add 'em together and we get 26 square units for the trapezium.

Here's another example, find the area of the trapezium by using the formula and the formula is provided there.

So to do that I need to first figure out which number is gonna go into which variable on my formula.

So the A and B represent the parallel sides, so that must be three and five.

It doesn't really matter which way round I put them 'cause I'm adding them together.

So that's really matter.

And the H is the height.

So the height is perpendicular to the parallel sides so that must be the four, and if you struggle to see why four must be the height, you can always turn your head sideways.

So either the five or the three is horizontal, you put those numbers in, I get half times the sum 3 and 5 multiply by 4, which gives 16 and its square units in this case.

Here's your example, find the area of this trapezium by using the formula and it's provided there for you.

Pause the video, have a go and press play when you're ready to go through the answer.

Okay, let's see how we got on.

We'll go to substitute four and seven for A and B, and then we'll substitute the six in for the H.

So we have a half times the sum of seven and four times by six to get 33 square units.

Okay, it's over to you now for task B, this task consists of two questions and here's question one.

The letters inside each shape represent the areas.

So T represents the area of the trapezium, A represents the area of the parallelogram.

Use each letter once to fill in the gaps in the calculations below.

So you can see it says T equals, and then you've got F plus something plus something and then T equals something plus something and so on.

You need to fill those blanks in with A, B, C, D, E, F, G, and H.

Now there are multiple different solutions, so what you get might be different as what someone else gets or once you've found one way you might wanna explore and see if you can find a different way.

Then once you've finished part A, have a go at part B.

That is find the value of T, which is the area of the trapezium, pause the video, have a go at this and press play when you're ready to continue.

And here is question two.

The three shapes below are all trapezia.

So you can be assured that you have a pair of parallel sides in each.

Now for trapezium with parallel sides A and B, the formula is provided there.

A equals half of the sum of A and B times by H.

Use that formula or any other method to calculate the area of each trapezium below.

Pause the video, have a go at these and press play when you're ready for some answers.

Okay, well done with that.

Let's now go through some answers.

With question one, there are multiple solutions, but here's one possible solution you could do T equals F plus B plus D.

Now that is a triangle plus a rectangle plus another triangle.

And then we can do T equals C plus E.

That is the parallelograms plus the triangle, and then T equals G plus H, which is a triangle plus another triangle.

Now be trapezium cut by its diagonal and that leaves T equals A divided by two.

That's the parallelogram divided by two.

So there's one possible solution.

I wonder if you can see why that parallelogram divided by two gives you the trapezium.

You could think of it as if you took two copies of that trapezium T, could you put those two copies together to make the parallelogram? You might have to turn one of those copies upside down and then put 'em together on the side that matches.

And you'll have something that looks a bit like parallelogram A.

A different solution could be something like this.

So you've got F plus C plus D, and you've got B plus E, G plus H, and then still A divided by two.

Now I wonder if you found any other solutions yourselves.

If you have and you wanna check them, you could find the area of each of the shapes individually and then check when you put into those calculations, you should get 15 for your answer each time.

Because find the value of T, it's 15 units squared.

And in question two, the area for the trapezium in A is 16 centimetres squared.

In B, it's 10 centimetres squared, and in C it's 15 centimetres squared.

Well done so far we're now we're onto the third and final learn cycle for this lesson, which is finding the areas of composite shapes.

Areas of composite shapes can be found by splitting them into shapes whose areas can be found.

So there's no form necessarily that's well known for finding the area in the L shape, but what we can do is split it into shapes that we've found the areas of so far.

We could split it into two rectangles like this, find the area of each rectangle, then add those together to get the area of the L shape, which is 36 units squared.

Or we can take this composite shape and split it in a different way.

We could split it from its diagonals and we'll have two trapezia.

If we find the area of each of those trapezia would have 11 square units and 25 square units.

So to find the area of the composite shape, we could combine those together to get the area of 36 square units.

So let's do an example together and then you'll do one similar yourselves.

Find the area of this composite shape.

Now, it's a composite shape, which means we probably won't have a formula to find the area of this whole shape.

So we need to do it by thinking about splitting it into shapes that we do have formulas for.

We could split it like this.

So we have a triangle and a rectangle.

If we look at those shapes separately, we've got a triangle which is six by four.

Now that four isn't on the original composite shape, so where's that four come from? Well, the height of the overall shape is seven, and the height of the rectangle part is three.

That must leave four left over for the height of the triangle.

So that's where the four comes from.

And then the rectangle is six by three.

If we find the areas of each of those shapes, we get 12 centimetres squared for the triangle, 18 centimetres squared for the rectangle.

And then to get the overall area of the composite shape, we can add those together to get 30 centimetres squared.

Okay, here's one for you to try.

It's a composite shape.

You need to find the area.

I recommend looking for shapes that you recognise within the composite shape.

Can you split this into two or more shapes that you know how to find the areas for? Pause the video while you have a go at this and press play when you're ready to go through it together.

Okay, one way we could split this shape is into a triangle and a parallelogram.

We look at these separately.

Our triangle is four by four and then the parallelogram is four by three.

Once again, that three is not on the original diagram.

We had to work out by doing seven, subtract the four to get the height of three.

Find the areas, add 'em together, and we get 20 centimetres squared.

How do we get on with that? Okay, here's another example to go through and this one has a slightly different technique.

Find the areas composite shape.

Now we could split this into smaller shapes and add 'em together.

For example, we could split it into two trapezia or we could try a different strategy.

We could fill in the empty space at the top there and consider what we might do with that.

So we fill in the empty space would have a rectangle and that empty space itself, we can see there is a triangle.

The measurements on here for the rectangle, we've got five by six, we can see that.

For the triangle, we've got six going across the top.

And then we've got two for the height there because that's what's left over from the five when we subtract the three.

If we find the area of each of those, we've got the rectangle has an area of 30 metres squared.

Now that's more than the overall shape because remember we filled a bit of space in.

And we have a triangle with the area of six metres squared.

Now this time to find the area of the composite shape, we're not gonna add these together.

It's like we've cut the triangle outta the rectangle to create the composite shape.

So this time we're gonna do 30, subtract 6 to get 24 metres squared.

So this strategy is about filling in empty space and then subtracting it afterwards.

So, here's one for you to try.

Pause the video, have a go at this and press play when you're ready to work through the answer.

So it looks like we have a triangle with a rectangle cut out of the bottom of it.

So let's fill that space in.

We've got a triangle, which is six by four, and then a rectangle, which is two by one.

Get the areas each of those, and then to create the composite shape, we will cut the rectangle out of the triangle.

So do 12, subtract 2 to get 10 metres squared.

Okay, it's over to you now for task C.

This task contains one question and here it is, find the area of each composite shape.

Pause the video, have a go and press play when you're ready to continue and go through some answers.

Well done with that.

Let's go through some answers.

Part A, the answer is 35 centimetres squared.

Part B, the answer is 23 millimetres squared and part C is 18 centimetres squared.

Fantastic work today.

Let's now summarise what we've learned in this lesson.

The area is the size of the surface and it states the number of square units, which is needed to completely cover that surface.

And a formula can be used to find the areas of rectangles, triangles, and parallelograms. There's also lots of different ways you can find the areas of trapezium, but there's also a formula as well.

But you can do it by splitting it into other shapes if you don't remember the formula.

And the areas of composite shapes can be found by splitting them into shapes which areas can be found.

Now what, whether that is splitting it into smaller shapes, find the areas and add 'em together, or whether it's filling in empty space, finding the areas and doing a subtraction both those methods are available.

Well done today.