Lesson video

Hello, my name is Dr.

Rowlandson and I'll be helping you with your learning during today's lesson.

Let's get started.

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

This lesson is called area of a circle, and by end of today's lesson, we'll understand the derivation of the area of a circle.

Now here are some keywords from previous lessons that we'll be reusing a lot in today's lesson.

So you may wanna pause the video while you remind yourself what these keywords mean before pressing play to continue.

This lesson contains two learn cycles.

In the first learn cycle, we're gonna start with estimating areas of circles to get a sense of how big they are.

And in the second learn cycle, we're gonna derive a formula to calculate the areas of circles in a much more accurate way.

But let's start off with estimating areas of circles.

Now here we have Laura.

Laura is estimating the area of the circle.

She says if I overlay the circle onto a grid, then I could estimate the area by counting unit squares.

So she does that.

Here we now have a circle over a grid, and each squareness grid has a length of one unit.

So each is a unit square.

She says there are 16 squares in total, so we can write down the total number of squares is 16.

She says the circle takes up less space than that, so the area of the circle must be less than 16 square units.

Hmm.

She then says there are four complete squares inside the circle, and we can see that 'cause they are highlighted there.

She says, well, the circle takes up more space than that, so its area must be greater than four square units.

So we now have that the total number of squares is equal to 16.

The number of squares inside is equal to four.

So the area of the circle must be between four and 16 square units.

And we've got it written there as an inequality at the bottom.

The area of the circle is greater than four square units, but it's less than 16 square units.

Let's check how well we've understood that so far.

Here we have a circle which is overlaid over a grid.

How many unit squares are there in total in this diagram? Pause the video while you have a go and press play when you're ready for an answer.

The answer is 25 unit squares in total.

So how many complete unit squares are inside the circle? Pause the video while you have a go at this and press play when you're ready for an answer.

Now there are some squares at the top, bottom on the left and right that look like they might be complete, but they're not quite complete.

We have nine complete unit squares inside this diagram.

So in that case, which range would the area of the circle be in? You got options A, B, and C there.

Pause the video, make a choice, and press play when you're ready to continue.

Down to this one is B.

We know that the area of the circle must be more than the nine squares that are complete inside it, but less than the 25 squares that are in total overlaid over it.

Here we have Jun.

Jun is using reasoning to think about the area of a circle.

He says areas are normally calculated by multiplying two perpendicular lengths, but I can only see one length on the circle.

That's the radius.

But he then says the radius in a circle is the same in every direction.

So I could draw on another radius that is perpendicular to the first one like this.

If I multiply the two radii, I would have the area of a square whose length is equal to radius.

So we multiply those together, we'll get the area of that square that's just appeared there.

If we just look at that in isolation now, we've got a square where each length is equal to radius.

And if we multiply R by R, we get R squared.

So the area of this square will be equal to the radius squared.

Hmm.

He then looks back at his circle and thinks, well, the area of the circle is greater than the area of one of these squares.

So we can write that down as area of a circle is greater than R squared.

He then says, well, the area of the circle looks convincingly greater than the area of two of these squares.

So we've got written the area of the circle is greater than two R squared.

He says, well, it's difficult to see whether the area of the circle is less or greater than the area of three squares.

That's not quite as straightforward to see whether all those extra bits around the outside add up to the quarter of the circle that's left inside.

So he says, well, the circle is fully inside four squares, so its area is definitely less than four squares.

So now we have an inequality that says the area of a circle is greater than two R squared, but less than four R squared.

So the area of the circle must be somewhere between two and four lots of the radius squared.

And we've got that inequality there.

It says the area of the circle is greater than two R squared, but less than four R squared.

Jun uses his conclusion to estimate the area of this circle.

We have a circle here with a radius of six metres.

He starts by thinking first about the square that has the same length as the radius.

That square be six by six.

So the area of that square would be 36 metres squared.

He then thinks about how the area of the circle is somewhere between two and four times the area of the square.

So while the area of two squares is 72 centimetres, the area of four squares is 144 centimetres, which means the area of the circle must be somewhere between 72 and 144 centimetres, which is written as an inequality here, area of the circle is greater than 72 centimetres squared, but less than 144 centimetres squared.

Let's check how well we've understood that.

Here we have a diagram with a circle and a square, and we can see that the height of the square is equal to the radius of the circle, and the diagram tells us that the area of the square is 11 centimetres squared.

And we also know that the area of the circle is between two and four times the area of the square.

So our question is, which is the most appropriate range for the area of the circle? You've got options A, B, and C.

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

The answer is B.

The area of the circle must be between 22 and 44 centimetres squared.

So how about this one then? The area of the circle is somewhere between two and four times the area of the square.

This time, the length of the square is given as five centimetres, which you can see is also the radius.

Which is the most appropriate range for the area of the circle this time? Pause the video while you make a choice and press play when you're ready for an answer.

Well, this time we need to first find the area of the square, which is five squared, which gives you a 25 and then times it by two and times it by four to get the two ends of our range, which means the answer is C, the area of the circle must be between 50 and 100 centimetres squared.

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

This task contains two questions where you'll be using the methods we've used so far for estimating the areas of circles in each of these two questions.

In question one, we've got circles overlaid over a grid and you need to first find the number of complete squares inside the circle, the number of total squares altogether on the grid, and then complete the range for estimating the area of the circle.

Pause the video while have a go at this and then press play when you're ready for question two.

And here is question two.

Use the information provided to fill in the gaps.

So the area of each circle is somewhere between two and four times the area of its accompanying square.

You can see each time when you've got the square there, its length or its height is equal to the radius of the circle.

So in part A, B, and C, find the area of two squares, find the area of four squares and then use it to complete your inequality for the area of the circle.

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

Okay, well done with that.

Let's now go through answers to question one.

So in part A, the number of squares inside the circle is 16.

The number of squares altogether in that grid is 36.

So that means that the area of the circle must be greater than 16 units squared and less than 36 units squared.

And part B, the number of squares inside is 32, and then the number of squares altogether in that grid is 64, which means the area of the circle must be between 32 and 64 unit squared.

And then question two.

So in part A, the area of that square is given to fours.

It is nine metres squared, so the area of two squares will be 18 metres squared and the areas of four squares would be 36 metres squared.

So our inequality will be A is greater than 18, but less than 36.

In part B, this time we define the area of the square ourselves first, that's four squared, which makes 16.

So that means the area of two squares must be 32.

The area of four squares must be 64.

So the area of the circle must be greater than 32, but less than 64 metres squared.

And then part C, we define the area of the square, but the radius is given to us.

Well, that's okay because the height of the square or the length of the square is the same as the radius.

So the area of the square will be 10 squared, which is 100.

Areas of two squares would be 200.

Area of four squares would be 400.

So our inequality is C is greater than 200, but less than 400.

Well done so far.

Now let's move on to the second part of today's lesson, which is deriving a formula to calculate the areas of circles accurately.

Here we have Sam.

Sam uses a computer to draw lots of circles and measure the area.

For example, Sam might be using a piece of dynamic geometry software.

Here's Sam's first circle.

It's got a radius of one and the computer calculates the area to be 3.

14159, and there are more decimals after that.

When the radius is two, the area is 12.

56637 and more decimals.

When the radius is three, the area is 28.

27433 and more decimals.

When the radius is four, the area is 50.

26548 and more decimals.

And when the radius is five, the area is 78.

53981 and more decimals.

Looking at this, Sam says, I'm struggling to spot a pattern in the numbers.

And you can't blame Sam because the differences each time are different.

There's not a constant difference between all those answers.

Add 9.

42, add 15.

70, and so on and so on.

But the number in the first row looks a bit like pi, 3.

14159 and more decimals.

Hmm.

Sam uses the computer to recalculate the areas, but this time in terms of pi.

Let's see what we get.

When the radius is one, the area is pi.

When the radius is two, the area is equal to four pi.

When the radius is three, the area is equal to nine pi.

And when the radius is four, can you guess what it might be? The area is equal to 16 pi.

And then when the radius is five, can you guess what this might be? The area is equal to 25 pi.

Sam now says I can see a relationship between the radius and the area of the circle.

Can you see it too? The area of the square is five squared, which is 25.

The area of the circle is then that answer multiplied by pi.

Is that the case for all of them? Well, yes it is.

When the radius is four, the area of the square would be 16 and then the area of the circle is 16 pi.

If we have the radius of three, if you square that, you get nine and then the area of the circle is nine pi.

So if the area of the circle is equal to pi times the area of that square we can see with the lengths equal to radius, then we can generalise what we've got here.

The area of the square is equal to the radius squared, so we can put that as R squared.

We then multiply by pi to get the area of the circle, so we get pi R squared.

So whatever the radius is, if we square it, we get the area of that square we can see and we times it by pi, we get the area of the circle.

So let's check what we've understood there.

The length of the square here is equal to the radius of the circle.

Can you fill in the blank here? Area of the circle is equal to something times the area of the square.

What is that something? Pause the video while you write it down, and press play when you're ready for an answer.

The answer is the area of the circle is equal to pi times the area of the square.

Now the length of the square is equal to the radius of the circle.

Again, the area of the square this time is 10 centimetres.

So what is the area of the circle? Pause the video while you have a go at this and press play when you're ready for an answer.

If the area of the square is 10 centimetres squared and the area of the circle is pi times that, then we would have an area of 10 pi centimetres squared.

You may have put it to decimal, which would be 31.

4159 and so on, depending on where you rounded it.

The length of the square is equal to the radius of the circle.

Again, in this question, the length of the square is nine centimetres.

What is the area of the circle? Your choices are A three pi centimetre squared, B nine pi centimetre squared, or D 81 pi centimetres squared.

Pause the video while you have a go at this and press play when you're ready for an answer.

The answer is C 81 pi centimetres squared.

We get that by taking the length of that square, which remember, it's equal to the radius of the circle.

If we square it, we get 81, which is the area of the square and times it by pi and we get 81 pi, which is the area of the circle.

And one more.

The area of the circle here is nine pi centimetres squared.

In which square is the length equal to a circle's radius? For options A, B, and C, pause the video, make a choice, and press play when you're ready for an answer.

The answer is B.

And the reason why is if the area of that square is nine, if we multiply it by pi, we get nine pi, which is the area of the circle.

That means the length of that square must be equal to the radius of that circle.

In other words, the radius of circle is three.

Let's now take a look at this formula in a bit more detail to try and understand why it works for finding the areas of circles.

So the formula A equals pi R squared can also be derived by splitting a circle into sectors.

Let's do that together now.

Let's take this circle and split it into four sectors and then if we take each of other sectors as pieces and rearrange them into another shape, let's see what we get.

So here's our first sector and then here's the next one and the next one and the next one.

Let's take a look at our new shape.

If we look at the circle on the left and you can see where the radius is in that circle, it goes in every single direction from the centre to the edge, can you see where our new diagram would be equal to the radius? There's quite a few options, but one interesting place is from here, from the bottom of the curve to the vertex, that is equal to the radius.

Let's now think about the circumference of the circle.

The circumference is equal to the distance around the edge of the circle, or in other words, it is equal to the distance around the curved edges of each of these four sectors.

And the formula for circumference is two times pi times the radius.

If we just had half a circumference then, that would be the distance around the edges of two of those sectors and that would be equal to pi times just the radius 'cause it's half of two pi R, which means on our new diagram, if we look at the curved edges along the bottom of it, well, that's two sectors and that means that distance along the bottom of our new shape is equal to half of the circumference of the circle, therefore it's equal to pi times the radius.

Now each time we split the circle into more sectors, the new shape tends more and more towards being a parallelogram.

For example, here I have six sectors.

If I piece those together, I get something to be like this.

Now this looks a little bit more like a parallelogram more than last time.

And if we look at where the radius is, it goes from the bottom of that shape to the top of that vertex.

Again, half of the circumference is equal to pi R, which means the distance across the bottom of our new shape is equal to half of the circumference, which is also pi R.

If we split into more sectors, we've got eight sectors here.

Let's look at that.

This is looking even more like a parallelogram.

And again, our radius goes from the bottom of our new shape to the nearly the top of it, that vertex there.

The distance across the bottom of our circle across half of the circumference is equal to pi R.

And that's the same as the distance across the bottom of our new shape.

So the bottom of the new shape, the base of it is equal to pi R as well.

Let's split into more sectors.

We get some for looks even more like a parallelogram.

The height of this new shape is equal to the radius or pretty much equal to the radius, and the base of our new shape is equal to half of the circumference again.

So the base is pi R and the height is approximately R, and then we split into more sectors.

It looks even more like a parallelogram and those curvy sides are getting straighter and straighter every single time until we get something that looks pretty much like a parallelogram.

Now I've stopped here with this many sectors, but imagine splitting it into a million sectors or a billion sectors and how much like a parallelogram that would look, the more we do, the straighter those sides become and more upright it becomes as well.

So the area of the parallelogram on the right-hand side is equal to base times the height.

And we can see that the base is equal to pi R because it's half the circumference and the height is equal to R.

So the area must be equal to pi R times R.

In other words, it's equal to pi R squared.

And we've seen that formula before.

That's because the area of the circle is equal to pi times the radius squared, which is A equals pi R squared.

So there we have derived the formula for the area of a circle by splitting it into lots of sectors, piece 'em together to make a parallelogram, and use our formula for the area of parallelogram to calculate the area of the circle.

And that is area equals pi times the radius squared.

Let's use this now.

The area of this parallelogram is equal to the area of the circle.

Find the area of each shape.

So we can do either of them.

Let's look at the parallelogram on the left.

We know the formula for the area of a parallelogram is base times height.

So let's substitute our values in.

Our base is three pi, our height is three, so the area of that parallelogram is equal to nine pi centimetres squared.

Now we know that the area of the circle is equal to that, but let's use our formula to double check it.

Area of the circle is equal to pi times the radius squared.

So we'll do pi times three squared 'cause three is the radius and we get nine pi again centimetres squared.

So let's break this down a tiny bit more.

The area of the circle is equal to the area of the parallelogram.

If the radius of the circle is eight centimetres, let's think what else we know.

What is the height of the parallelogram? The parallelogram must also be eight centimetres because the height of this parallelogram is equal to the radius of the circle.

What is the circumference of the circle? Well, to get a circumference, we would do eight times two times pi, or we could get the diameter 16 and times that by pi.

But either way, we get 16 pi centimetres.

So if that's a circumference of the circle, what is the base of the parallelogram? Now the base of the parallelogram is equal to half of the circumference of the circle.

So the base of parallelogram must be equal to eight pi centimetres.

What is the area of each shape? Well, the area of the parallelogram is equal to base times height, which is eight pi times eight, which is 64 pi centimetres squared.

And the area of the circle is pi times radius squared.

So that's pi times eight squared, which is also 64 pi centimetres squared.

Let's check what we've understood there.

So here we have a circle and a parallelogram and we're told that the area of the circle is equal to the area of the parallelogram.

What is the height of the parallelogram? Pause the video, write down your answer, and press play when you're ready to continue.

The height of the parallelogram must be six centimetres.

What is the circumference of the circle in terms of pi? Pause the video, write down your answer, and press play when you're ready for an answer.

The circumference of the circle must be 12 pi centimetres because it's two times six times pi.

So in that case, how long is the base of the parallelogram? Pause the video, write down your answer, and press play when you're ready to continue.

The base of the parallelogram is half of the circumference, so it must be six pi centimetres.

So what is the area of each shape in terms of pi? Pause the video while you have a go and press play when you're ready for an answer.

The area of each shape must be 36 pi centimetres squared, whether you do that on the parallelogram by doing six pi times six, or do it on the circle by doing six squared times pi.

Either way, you get 36 pi centimetres squared.

Okay, it's O to you now for task B.

This task contains two questions and here is question one.

Each circle overlaps with a square, and for each question find the area of the square and then find the area of the circle and give your answer in terms of pi.

Pause the video while you have a go and press play when you're ready for question two.

And here is question two.

We've got three parallelograms and three circles underneath them.

The question says, each parallelogram is made from the sectors of one of the three circles.

Now the diagrams are not drawn to scale, so we can't do this by measuring with a ruler.

You need to match each parallelogram with a circle that has the same approximate area to begin with.

And then once you've got that, find the area of each circle in terms of pi.

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

Okay, well done with that.

Let's now work through these questions together.

In question one, in part A, the area of the square, it's nine times nine, which is 81 centimetres squared.

So if we times that by pi, we get the area of the circle, which is 81 pi centimetres squared.

In part B, the square has a length of four centimetres, so its area must be 16 centimetres squared, which means the area must be 16 pi centimetres squared for the circle.

And in part C, we don't have the length of the square, but we do have a radius.

But we can see that the length of the square is equal to the radius.

So the square must be seven by seven, which is 49 centimetres squared and times it by pi, we get 49 pi centimetres squared for the area of a circle.

And in question two, let's start by matching these up.

So in the first one, we can see that the first circle on the left matches with the middle parallelogram.

That's because the radius of that first circle is 20 and the height of that parallelogram is also 20.

The second circle matches up with the parallelogram on the far right.

Now, if the circumference of the circle is 10 pi, the base of the parallelogram is half of that, which is five pi, and that leaves us with the third circle matches the first parallelogram.

Let's double check that.

The diameter of the circle is 20 metres, which means the radius must be 10 metres.

And we can see the height of the parallelogram is 10 metres.

We can also double check it by thinking if the diameter is 20, the circumference must be 20 pi.

And if you're half that, you get 10 pi, which the base of the parallelogram.

Now we've got that.

Let's find the area of each circle in terms of pi.

We can do it either by finding area of the circle using the formula or find the area of the parallelogram using the formula of that one instead.

Either way, we get 400 pi metres squared, 25 pi metres squared, and 100 pi metres squared.

How do we go on with that? Great job today.

Absolutely fantastic work.

Here's a summary of what we've learned in this lesson.

The area of a circle can be estimated by using squares, and you can use those squares in different ways to do that estimation.

However, the main focus of today's lesson has been about deriving a formula to calculate the area base circle exactly.

Now a circle can be cut into sectors that are then placed together to make a parallelogram.

And the more sectors we cut it into, the more that new shape looks like a parallelogram.

Now the length of the parallelogram is equal to half of the circumference, and the height of the parallelogram is the radius of the circle.

So the form of the area of the circle can be derived from this parallelogram, base times height, half the circumference times the height or pi R times R gives us the area of the circle, which is pi R squared.

Great job today.

Well done.