Lesson video

Hello, my name is Dr.

Rowlandson, and I'm thrilled that you're joining me in today's lesson.

Let's get started.

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

This lesson is called Multiplicative Relationships in Circles.

And by the end of today's lesson, we'll be able to recognise that there is a constant multiplicative relationship between the diameter and the circumference of a circle, and it's called pi.

This lesson will be introducing three new key words.

Now, we'll be defining each of these words in more detail later in the lesson, but feel free to pause the video and have a read of the definitions now before we start.

This lesson contains three learning cycles.

In the first learning cycle, we'll be learning the names of the parts of a circle.

In the second learning cycle, we'll be learning about the relationship between the circumference and the diameter.

In the third learning cycle, we'll be deriving a formula to calculate the circumference of a circle.

But let's start off with learning the parts of a circle.

A circle can be described based on its parts.

Here we have a circle.

The circumference of a circle is the distance around its edge.

In other words, how far is it along that purple curved line on that circle there? The circumference is also a word used for perimeter of curved shapes in general.

For example, on the right-hand side, that's not a circle, but we can describe that distance around its edge as the circumference as well.

A circle is a shape such that every point on the circumference is equidistant to its centre.

So here we have a circle, and we can see that the distance from the centre of the circle going upwards to the circumference is equal to a, and also, the distance going from the centre of the circle right to the circumference is equal to b.

a and b must be equal to each other because every point on that circumference is equidistant to the centre.

And no matter which direction a and b go in, a and b will always be equal to each other.

Whereas on the right we have a shape that is not a circle.

And we can see there that the distance from the centre going upwards, a, is longer than the distance from the centre going rightwards, which is b.

Therefore, a and b are not equal to each other, so it's not a circle.

The distance from the edge of a circle to its centre is called the radius.

So this purple line segment here we can see goes from the edge to the centre, or from the centre to the edge.

Therefore, it is the radius of the circle.

And the radius of a circle is any line segment that joins the centre to its edge.

And its length is the same in every direction.

So that's a radius, that's a radius.

And the plural for radius is radii.

In these non-examples, none of these purple line segments are a radius.

This one is not a radius because it goes from an edge of the circle to another part of the edge.

This one's not a radius because yes, it goes from the centre, but it doesn't go all the way to the edge.

And this is not a radius because it goes from the edge, but it doesn't get to the centre.

So it's not a radius.

The greatest distance that can be travelled from one side of a circle to another must go through the centre of the circle.

And its length is called the diameter.

So the diameter of a circle is any line segment that starts and ends on the circumference of a circle and passes through the centre.

So that line segment we can see there is a diameter.

So is this one, and so is this one.

In these non-examples, neither of these line segments are the diameter.

This one is not a diameter because it does not go through the centre.

And this one's not a diameter because it doesn't go all the way from one side of the circle to the other through the centre.

Alex and Izzy are exercising on a field with a marked circle.

Alex runs from the edge of the circle to its centre and then out to another point on its edge.

And Izzy runs in a straight line from one side of the circle, through its centre, to the other.

So Izzy runs one hmm of the circle.

What word goes in that blank space based on the words we've learned so far? Izzy runs one diameter of the circle, 'cause she goes from one side to the other, through the centre.

Alex runs two hmm of the circle.

What's the word that goes in that blank there for the two things that Alex runs? Well, he runs from the edge to the centre and then the centre to the edge.

Each of those lines, line segments, are a radius.

So Alex runs two radii of the circle.

Who runs the greatest distance? Pause the video while you think about this, and press play when you're ready to continue.

They both in fact run the same distance.

If you look at where Alex runs, he runs from the edge to the centre to the edge.

Now, the distance from the centre to the edge is the same in every direction, so Alex's distance would be the same if he ran like that, from the edge to the centre to the edge, in that way.

And we can see more clearly now that the distance that Alex runs must be the same as the distance that Izzy runs, because it's from one side of the circle, through the centre, to the edge.

Like that.

So the diameter of a circle is twice the length of the radius.

Here we can see we've got the diameter, goes from one side of the circle to the other, and it goes through the centre.

Whereas the radius, each radius goes from the edge to the centre.

So if we have two of them going in a straight line, they'll be equal to the diameter.

So diameter of a circle is twice the length of the radius.

Or we can even say that the radius of a circle is half the length of the diameter.

Like that.

For example, if the radius has length five, then each radii also has length five, which means the diameter must have length 10.

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

Which part of the circle is labelled? Is it a, the circumference, b, the diameter, or c, the radius? Pause the video while you have a go, and press play when you're ready for an answer.

The answer is c, it's the radius.

It's the length from the centre to the edge of a circle.

Fill in the blank.

The hmm of a circle is the distance around its outside.

Pause the video while you write the word that goes in that blank, and press play when you're ready for an answer.

The answer is circumference.

The circumference of a circle is the distance around its outside, very similar to the perimeter of a shape.

And another question.

The radius of the circle is eight metres.

How long is the diameter? Pause the video while you have a go at this, and press play when you're ready for an answer.

Answer is 16 metres.

The diameter is twice the length of the radius.

So the radius is eight.

8 times 2 is 16.

The diameter of the circle is eight metres.

How long is the radius? Pause the video and have a go, and press play when you're ready for an answer.

This answer is four metres.

The diameter is twice the radius, or you can think of it as the radius is half of diameter.

Half of eight is four metres.

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

This task contains two questions, and here is question one.

For each diagram, explain why the length of the line segment is not the radius.

You'll have to write a sentence or two for each question to explain why it's not the radius.

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

And here is question two.

Fill in these blanks with numbers.

So you've got five statements there for parts a, b, c, and d.

Read each one and replace the blank with a number.

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

Okay, great job with that.

Let's now go through some answers.

Question one, we need to explain why each line segment is not the radius.

So in part a, the line segment does not go through the centre of the circle.

Now you might have worded that answer differently, but you've gotta highlight the key point there, which is doesn't go to the centre.

In part b, the line segment does not go to the edge of the circle.

And in part c, yes, it goes to the edge of the circle, but it keeps going.

So that line segment goes beyond the edge of the circle.

So that is more than the radius in that case.

In question two, you need to fill in the blanks of numbers.

The diameter of a circle is two times the radius.

Part b, a circle with a radius of six centimetres has a diameter of 12 centimetres because it's twice the radius.

In part c, a circle with a diameter of six centimetres has a radius of three centimetres because it's half of the diameter.

In part d, the distance from the centre of a circle to its edge is 10 metres, so its radius is 10 metres, and its diameter is 20 metres.

And part e, the furthest distance between two points on the edge of a circle is 24 metres.

That's the diameter.

Its radius is 12 metres, and its diameter is 24 metres.

How did we get on with that? Great work so far.

Now let's explore the relationship between the circumference and the diameter of a circle.

Aisha is comparing the circumference of a circle with its diameter.

And she starts by just looking at half of the circle like this.

And she says, "The curved distance is further than the straight distance.

Therefore, half of the circumference must be greater than the diameter." We can see that the curved distance is half of the circumference.

It's half the distance around the edge of the circle.

And the straight line going across the bottom, well, that's the diameter, because it goes from one side of the circle, through the centre, to the other side.

And we can see that, very visually, that the curved line is longer than the straight line.

So that's why Aisha's reasoned that half of the circumference must be greater than the diameter.

That means that the whole circumference must be greater than two diameters.

Because if we go all the way around a circle, that's two of those curved lines, which is the whole circumference.

And that is further than going from one side of the circle to the other and then back again, which is two diameters.

She then says, "If we draw a square around the circle, then each of its lengths will be equal to the diameter." We can see that if we draw the diameter horizontally through that circle, the length at the top of the square and the bottom of the square are both equal to that diameter.

If we draw the diameter going vertically up through the circle, then we've got two vertical lengths on the square, on the left and the right.

They must be equal to the diameter as well.

So each of those four lengths on the square are equal to the diameter.

Therefore the perimeter of the square must be equal to four diameters.

Now, the distance around the circle is less than the distance around the square.

So that means the circumference must be less than four diameters.

So put out together, Aisha thinks, "Well, the circumference is more than two diameters, but it's less than four diameters.

So the circumference must be around three times the diameter.

I wonder if it's exactly three diameters, or if the multiplier might be a decimal." Maybe it might be three point something times the diameter, or two point something times the diameter.

Let's investigate this further.

So Alex is exploring the relationship between the diameter and circumference of a circle in a practical way.

And he's doing it using a roll of toilet paper.

And you could try this yourself as you want to as well.

The cross section of the roll of toilet paper is pretty much a circle.

It's not perfectly a circle, you can see there's a few bumps in there, but it's pretty much a circle, which means the distance around the edge of the toilet roll is equal to the circumference, and the distance from one side of the toilet roll to the other, going through the centre, is the diameter.

Alex says, "How large is the circumference of this toilet roll compared it to its diameter?" So he's gonna talk us through what he does.

I've marked where the end of the paper came to on the roll.

Can see you got your black felt-tip line running down a toilet roll there.

He says, "I'm unrolling the paper up to the mark, and then I'll cut the paper along the mark." He then says, "The length of the paper that is cut off is approximately equal to the circumference of the roll." He says, "I've placed the roll over the paper at the end, and I've marked where the roll comes to on the paper." You can see that.

And the space between these marks is approximately equal to the diameter of the roll, 'cause that was a distance from one side of the roll to the other.

He says, "I've placed the roll on the paper again and marked where it comes to this time.

And I did it again for a third time.

And then, hmm, there's not enough room left on the paper to do it a fourth time." So we've got the diameter and another diameter and another diameter and just a little bit of a diameter, but not a whole fourth diameter.

So the circumference was approximate equal to three and a bit diameters.

Izzy explores this in a different way.

She uses a computer to draw circles and measure the circumference and diameter.

For example, she might use a piece of dynamic geometry software for this.

So here's her first example.

The diameter is one, and the computer has measured the circumference as being 3.

141592654.

Her second circle looks like this.

Diameter of two, and the computer has measured the circumference as being 6.

283185307.

And then when the diameter is three, the circumference is 9.

424777961.

And when the diameter is four, the circumference is 12.

56637061.

And when the diameter is five, circumference is 15.

70796327.

Whew, those are long decimals.

Izzy then says that the values for the circumference in this table have been rounded.

The numbers that the computer showed had a lot more digits than these.

So while these are very long decimals, Izzy's saying that, actually, the computer gave a lot more decimals than that.

She says the circumference looks like it's increasing by a constant difference.

Let's take a look at that.

Izzy plots her measurements on a graph to explore relationships between the diameter and circumference further.

So here we've got an axis with diameter going horizontally, an axis for circumference going vertically.

The first circle she plots is a diameter of one and a circumference of 3.

14 something.

And then you can see that first dot there, that first point on the graph in the bottom left.

Then each point afterwards going upwards are the other four circles.

A diameter of two, a diameter of three, and so on.

Now, if we look at these points together on the graph, it appears that these points form a straight line, which would suggest that they have a constant difference between them.

And if we work out that constant difference by doing a subtraction, that constant difference is always 3.

14.

Each circumference is 3.

14 something, something, something more than the previous one.

And we can even see that on the graph because that is the gradient.

Each time we go across one, we go up 3.

14 something, something, something.

So Izzy says, "It appears that there is a multiplicative relationship between the circumference and the diameter." It is going up by 3.

14 something every single time, starting from a diameter of one.

Then we can see that the circumference is always roughly equal to 3.

14 times the diameter.

1 times 3.

14 gives you 3.

14, 2 times 3.

14 gives you 6.

28, and so on.

But we must remember it's not exactly 3.

14.

There are more digits after that.

The ratio between the diameter and the circumference of a circle is therefore always the same.

The circumference is always approximately 3.

14 something, something times the diameter.

However, this number has more digits after that 3.

14, and let's remember that.

In fact, the number of decimal places is infinite, meaning that the digits never end.

Now, while this information is currently presented as a table, we could also present it as a ratio, the diameter to the circumference, 1 to 3.

1459, and so on.

And we can see that a circumference is always 3.

14 times the diameter.

Now, that 3.

14 number, which is the ratio between the diameter and the circumference, that number is so long but so special that it has its own symbol.

We can represent the entire of that number using the Greek letter pi.

And pi is approximately equal to 3.

141596.

I'm not gonna read them all out.

There's loads of numbers there.

But in fact, there's even more numbers than that.

Pi has an infinite number of decimal places, so it cannot be written accurately in any decimal form.

But it's approximately 3.

14159.

But the most accurate way to write down the number is just using the symbol pi.

When you write that symbol down, it represents the entire of pi, including all of its decimals, so it's accurate.

True or false? The ratio between the diameter and circumference is always the same for every circle.

I'd like you to choose true or false and one of the justifications underneath.

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

The answer is true, because the circumference is always approximately 3.

14 times the length of the diameter.

True or false? The number pi is exactly 3.

14.

Choose true or false and one of the justifications below.

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

The answer is false.

No, it's not.

That's because pi cannot be written accurately as a decimal because it has an infinite number of digits.

There are more digits after that one four.

And another question.

Which is the most accurate way of writing the ratio between the diameter and circumference of a circle? You've got four options to choose from there: a, b, c, and d.

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

And the answer is d, write the symbol pi.

Now, you might be thinking, "Oh my word, do I have to type in 3.

14159, so on, so on, every single time you wanna do a calculation with pi?" Well, good news is scientific calculators have a pi button, so you can perform calculations with pi accurately.

On this calculator here, we've got a pi symbol above the seven button.

Now, 'cause it's above the button, we've gotta first press shift and then press the seven button to enter the pi symbol.

Like that.

So when typing a calculation using the pi button, the calculator will often give an answer that is in terms of pi.

So we can see here on the calculator the person has typed in 5 times pi, and the answer the calculator's given you is 5 pi.

That answer is in terms of pi.

Now, calculators can convert answers from being in terms of pi to decimals.

On this calculator, you can do this by pressing the format button and then selecting decimal, and it'll change that answer into a decimal form, which is 15.

70796327.

Now, do remember there will be more digits after that.

Those are just the ones that the calculator can show you.

So for example, the diameter of the circle is 20 centimetres.

Find the circumference.

To find the circumference of this circle, we should think about the ratio between the diameter and circumference.

The circumference for any circle is equal to the diameter times pi.

So in this circle we've got a diameter of 20.

The circumference will be equal to 20 times pi.

So to calculate circumference, I'll take my diameter of 20 and I'll times it by pi.

Now, the most accurate way to write the answer to this is in terms of pi, which would be 20 pi centimetres.

I wanna stress that the pi is not a unit.

The pi is a number 3.

14, so on, so on.

The units here are centimetres.

So 20 pi is the number, and centimetres is the units.

So that'll be the answer in terms of pi.

20 pi centimetres.

Or we can write it as a decimal.

To write it as a decimal, it would require rounding it, though, to an appropriate degree of accuracy.

For example, three significant figures.

So if we use our calculator to convert it to a decimal, we would get something that rounds to approximately 62.

8 centimetres.

Here's another example.

The circumference of the circle is 20 centimetres.

This time, find the diameter.

Give your answer to two decimal places.

Let's start again by thinking about that ratio between the diameter and the circumference.

The circumference is always bigger than the diameter.

The diameter is always smaller than the circumference.

In fact, the circumference is equal to the diameter times pi, which must mean that the diameter is equal to a circumference divided by pi.

So this time, when I've got my circumference of 20 centimetres, I know that my diameter must be less than that.

It must be 20 divided by pi.

So to get my diameter, I can do 20 divided by pi, which is approximately 6.

37 centimetres once it's rounded.

Okay, let's check what we've learned there.

The diameter of the circle is 12 centimetres.

Find the circumference, and give your answer in terms of pi.

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

Okay, so the answer is 12 times pi gives you 12 pi centimetres.

So here's the same circle.

This time find the circumference and give your answer to one decimal place.

Pause your video while you have a go at this, and press play when you're ready to continue.

Well, we can type in 12 times pi again, but this time convert our answer to a decimal and round it to get 37.

7 centimetres.

The circumference of the circle is 12 centimetres.

Find the diameter.

Give your answer to three significant figures.

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

So this time we're gonna do 12 divided by pi to get 3.

82 centimetres, after it's been rounded to three significant figures.

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

This task contains two questions, and they're both visible here on the screen.

In question one, you need to fill in the blanks in the table.

The table has the radius in one row, it gives you the diameter in another row, and the circumference in another row.

And you can write your answer in terms of pi.

And in question two, you need to find the missing length in each circle, giving your answer to three significant figures.

Pause your video while you're having a go at this, and press play when you're ready to go through some answers.

Well done with that.

Let's now check our answers.

In question one, once you fill in the table, you should have these values here.

Pause the video while you check these against your own answers, and press play when you're ready to continue.

And then in question two, need to find a missing length in each circle, giving the answers to three significant figures.

In the first one, the diameter is 11.

So to get the circumference, we need to multiply it by pi to get 34.

6 centimetres, after it's been rounded.

And in b, if the circumference is 80, we need to divide it by pi to get 25.

5 millimetres, after it's been rounded.

Fantastic work so far.

Let's now move on to the final part of this lesson, which is deriving a formula to calculate the circumference.

Here we have Sam.

Sam is calculating the circumferences of circles.

Here we have a circle that has a diameter of 10.

And Sam is done all the same things that we've done here by thinking about the ratio between the diameter and the circumference to do 10 times pi to get a circumference of 10 pi.

Sam thinks, "I can replace the 10 with whatever the diameter is." We're always doing the same thing with the diameter every single time, so if the diameter was 12, I'll do 12 times pi to get a circumference of 12 pi.

And if the diameter was 14, I'll do 14 times pi to get a circumference of 14 pi.

When calculations start to feel a bit formulaic, it can sometimes be a bit of a clue that we can maybe perhaps derive a formula out this situation.

So Sam says, "I can make this into a formula by using a variable for the diameter." Because the diameter is varying between these questions.

So if we replace the diameter with a d, perhaps, for diameter and use that as a variable to represent the length of the diameter, then the circumference will be equal to d times pi, or d pi.

Then Sam thinks, "Well, we need to make sure it's in good algebraic form.

Pi is a number, so it should go before the variable in its algebraic term." So the circumference is equal to pi d, instead.

And then, finally, Sam says, "I can use the letter C as a variable as well to represent the circumference." And now we have a formula: C is equal to pi d, where C represents the circumference and d represents the diameter.

So the formula C equals pi d can be used to calculate the circumference when the diameter is known.

Let's represent this with a diagram.

Here we've got the diameter, which is d, the circumference, which is C, and to get from the diameter to circumference, we multiply by pi.

We could represent this in words by saying the circumference is equal to pi times whatever the diameter is.

Or we can represent it with a formula by saying C equals pi d, where C is the circumference and d is the diameter.

Now, we also know that the diameter is equal to two radii.

So we could use that fact to derive another formula to calculate circumference.

If we look at the bit in the middle where it says in words, circumference equals pi times diameter, well, the diameter is equal to two radii.

So we could replace the word diameter with two radii.

And the same in the formula as well.

C equals pi d.

Well, d is equal to 2r, so we can replace the d with 2r.

So the formula can be adapted to calculate the circumference when the radius is known.

Circumference equals 2 times pi times radius.

That two has come from two radii makes a diameter.

And in the diagram, we could take the radius and times it by 2 pi to get a circumference.

Or in the formula, we could replace it with C equals 2 pi r, where the two and the r are the 2r that is equal to the diameter.

So we have two formulae that we can use to calculate the circumference of a circle.

one using the diameter, C equals pi d, and the other using the radius, C equals 2 pi r.

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

Which two formulae are correct for calculating the circumference of a circle? Your options are a, b, c, and d.

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

The circumference can be calculated using either C equals pi d or C equals 2 pi r.

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

This task contains one question, and here it is.

You need to match each circle with two formulae that calculates its circumference.

You can draw lines from the circles to the formulae on the right for this.

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

Okay, well done with that.

Let's now go through some answers.

The first circle has a diameter of 10, which means it has a radius of five.

So the two formulae we can use there are C equals pi times 10, that's the diameter, or C equals 2 times pi times 5, 'cause 5 is the radius.

The next circle has a radius of 10.

So we can either use the formula C equals pi times 20, because 20 would be the diameter, or we can use C equals 2 times pi times 10, because 10's the radius.

And the bottom circle, you need to think a bit more carefully about this one.

That length goes all the way from the bottom to all the way to the top.

That's the furthest distance on that circle, so that must be the diameter.

So we can use the formula C equals pi times 5, 'cause 5 is the diameter, or C equals 2 times pi times 2.

5, 'cause 2.

5 is the radius.

Wonderful work today.

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

The diameter of a circle is double the length of the radius, and the length of the radius is always half of the diameter.

The ratio between the diameter and circumference is always the same for every circle.

No matter how big or small that circle is, the ratio between diameter and circumference is always equal to pi.

And it's written with that Greek symbol, the letter pi there.

Now, pi cannot be written accurately in decimal form because its decimals are infinite.

But it's approximately 3.

14159.

And the circumference of a circle can be calculated using the formula C equals pi d or C equals 2 pi r.

Well done today.