Lesson video

Hello, my name is Dr.

Rowlandson and I'm delighted that you'll be 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 circumference of a circle.

And by end of today's lesson, we'll be able to use the relationship, C equals pi d, to calculate unknown lengths in context involving the circumference of circles.

Here are some previous key words that you may be familiar with and will be using again in today's lesson.

So, you may wanna pause the video at this point and re-familiarize yourselves to these words before we continue.

This lesson contains two learn cycles.

In the first learn cycle, we'll be using a formula to calculate the circumference of a circle.

And in the second learn cycle, we're using that same formula, but this time, to calculate the diameter and the radius when the circumference is given.

But let's start off with using a formula to calculate the circumference of a circle.

The circumference of a circle can be calculated when the diameter is known by using the formula, C equals pi d, where C represents the circumference, d represents the diameter.

So, the formula is circumference equals pi times diameter.

Or we can calculate a circumference when the radius is known by using the formula, C equals 2pi r, where this time, r represents the radius.

These two formula give the same results, because the diameter is double the length of the radius.

So, whereas in one formula, we've got pi times d, if we substitute that d with 2r, with two radii, we'd have the formula, C equals 2pi r.

This time, the two and the r are separated by the pi.

That's just good algebraic form to put your numbers before your variables.

Now, calculations involved in circles can be performed accurately by using the pi button on a scientific calculator.

On this calculator here, the pi symbol is above the seven button.

So, to get to it, because this symbol is above the button, not on the button, we need to press shift before then pressing the seven button and that'll enter pi into the calculator like this, shift and then seven.

It may be in a different place depending on what calculator you are using.

Now, we can use the diameter and the formula, C equals pi d, to calculate a circumference of this circle.

So, we can see we've got a circle and that length goes from one side of the circle through the centre to the other side.

So, that must be the diameter, which is 12 centimetres.

So, let's substitute that value into our formula, C equals pi times 12.

Now, using this calculator, we would enter the buttons, one, two, for the 12, times shift, and then press the seven button which brings up pi.

And then, press the execute button to get our answer.

And our calculator says 12pi and our units are centimetres.

So, C, the circumference, equals 12pi centimetres.

Now, we can convert this answer to a decimal by pressing the format button on our calculator, and then selecting decimal.

Press the format button, it comes up with some options.

We choose decimal, and then it says 37.

69911184.

We've got it as a decimal here.

Now, we will need to round our answer to an appropriate degree of accuracy.

For example, the circumference is 37.

7 centimetres.

Now, the circumference isn't exactly 37.

7 centimetres.

It's been rounded, which means we've lost a little bit of accuracy along the way there.

Now, the more decimal places that we use in our answer, the more accurate our answer will be.

For example, this answer, 37.

699, is more accurate than our previous answer of 37.

7.

And we write more decimals, then the answer's even more accurate and more accurate again.

But no matter how many decimals I write, the only way to be completely exact is to give our answer in terms of pi, which will be 12pi centimetres.

Now, instead of using diameter, we could use the radius and use the formula, C equals 2pi r, to calculate the circumference of a circle instead.

So, here, we've got a circle and the distance from the edge to the centre is six centimetres.

That means the radius must be six centimetres.

So, let's substitute that into our formula, C equals 2pi r.

So, we'd have C equals two times pi times six, which gives 12pi centimetres.

If we compare that to the calculations using the diameter, we can see we get the same result each time.

It gives the same results as using the diameter, because the diameter is double the length of the radius.

And we can see in the left-hand calculation, we have two and a six and they're being multiplied together.

On the right-hand calculation, we have a 12.

That is the answer to two times six.

So, in either case, we're doing 12 times pi.

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

True or false, you can only calculate a circumference when we know the diameter.

Choose either true or false and one of the justifications below.

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

The answer is false, because we can use the radius to calculate the circumference by using the formula, C equals 2pi r.

True or false? When multiplying a number by pi, the only way to provide an exact answer is to write it in terms of pi.

Choose true or false and one of the justifications below.

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

The answer is true, because pi has an infinite number of decimals.

So, writing it in terms of pi is the only way to be exact.

Okay, let's now work through an example together, and then you'll have a very similar one to try yourself in a few minutes' time.

Find a circumference of the circle.

Give your answer to one decimal place.

Now, that distance of 26 millimetres goes from one side of the circle to the other.

It's the furthest distance across that circle, which means it must be equal to the diameter.

The diameter is 26 millimetres.

So, if we have the diameter, we can use a formula, C equals pi d.

Substitute 26 in and we get C equals pi times 26.

And because we're working with decimals, we would ask our calculator to give a decimal answer.

It'll say 81.

6814, and a lot more decimals.

We need to round it to one decimal place then.

So, we'll get C equals 81.

7 millimetres given to one decimal place.

Here's one for you to try.

Find a circumference of the circle.

Give your answer to one decimal place.

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

Okay, so in this one, we've got the diameter again of four metres.

So, we need to use the formula, C equals pi d, substitute it in, and we get C equals pi times four, which is 12.

56637, and so on.

Round it to one decimal place and we get 12.

6 metres to one decimal place.

Here's another example.

Find circumference of the circle.

Give in your answer this time in terms of pi.

Now, we look at this dash square we've got in here.

We can see that the length goes from the centre of the circle to its edge, which means that eight centimetres must be equals to the radius.

So, we'll use the formula, C equals 2pi r, this time.

Substitute our numbers in.

We get C equals two times pi times eight.

We get our answer in terms of pi as 16pi centimetres.

Now, also remember that pi is not a unit, pi is a number, so there's 16pi.

Part of that answer is the number and the centimetres is the units.

So, you must always write centimetres even when working in terms of pi.

So, here's your question to have a go at.

Find the circumference of the circle.

Give your answer in terms of pi.

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

Okay, so that five metres is the distance from the centre of the circle to the outside, which is the radius.

Therefore, we're gonna use a formula, C equals 2pi r, substitute our numbers in, and we get 10pi metres.

Okay, over to you now for task A.

This task contains two questions and here is the first question.

Calculate the circumference of each circle.

Now, in each question, give your answer first in terms of pi, and then secondly, give it rounded to three significant figures.

So, each time, to give your answer twice.

Once in terms of pi, once is a decimal.

But make sure when you're rounding it, you're rounding correctly.

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

Question two, calculate the circumference of each circle and this time, give your answer rounded to one decimal place.

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

Okay, well done with that.

Let's now go through some answers.

Here is question one.

In part A, the diameter is nine.

So, our circumference is either 9pi centimetres or 28.

3 centimetres when given to three figures.

In part B, well, our radius is five, so our circumference will either be 10pi centimetres or 31.

4 centimetres to three significant figures.

In part C, that length there is a radius, because it's the length from the centre to the edge.

So, our circumference would either be 11pi centimetres or 34.

6 centimetres.

And in part D, that length is a diameter.

So, the circumference would either be 12pi centimetres or written as 37.

7 centimetres to three significant figures.

And in question two, you need to think a bit more carefully about what numbers you're gonna use in your formula.

So, in part A, the 14 is neither the radius or the diameter, but the 22.

8 is the diameter.

So, we substitute that in to a formula, C equals pi d, and we get 71.

6 metres.

Now, in part B, the 17 is neither the diameter or the radius, but the 11.

4, that is the radius.

So, if we substitute that into the formula, C equals 2pi r, we'd get 71.

6 metres.

Now, questions A and B, they have the same answer.

Can we see why? In part A, a diameter, 22.

8, well, that's double, 11.

4, which is the radius of part B.

And remember, the diameter is double the radius.

So, actually, part A and part B, those are the same circles.

And part C, this time, we've got 4.

1 metres.

Now, that actually is the radius that goes from the edge to the centre.

And that 8.

2 metres, well, that's the diameter, 'cause it goes from one side of the circle to the other through the centre.

So, we could use either of those numbers in either the formulas that matched them.

And look also, we can see that 4.

1 is half of 8.

2, so that works as well.

Whichever way we do it, we get 25.

8 metres.

And in part D, we need to figure out what either the radius or diameter is first.

We can see that the length of one of those squares is one metre.

So, if we look from left to right, we can see that the diameter is four metres or you can go from the centre to the outside and see it's two metres.

Either way, you should get 12.

6 metres as your circumference.

So far, so good.

Now, let's move on to this second learn cycle for today's lesson, which is using the formula to calculate the diameter or the radius.

In cases where the circumference is known, the formulae, C equals pi d and C equals 2pi r, can be used to calculate the diameter or the radius.

And we can do this by substituting the known values into a formula to form an equation, and then rearranging the equation to find either the diameter or radius.

Now, this will usually involve dividing circumference by pi or 2pi, because that's what's in each of those formula.

Now, dividing by pi or 2pi is likely to give a decimal answer.

So, decimal answers may need to be rounded to a suitable degree of accuracy.

So, let's do an example together now, and then you'll have a similar one to try yourselves.

The circumference of this circle is 20 centimetres.

Calculate its diameter.

So, because I'm calculating the diameter and I use the formula, C equals pi d, because that formula has the diameter in.

That's what I want.

Let's now substitute the 20 in for the C.

So, you get 20 equals pi d.

Now, the pi and the d are multiplying each other.

So, if I want to work out what d is, I'm going to divide both sides of my equation by pi.

On the right-hand side, the two pis will cancel each other out.

On the right-hand side, I've got pi divided by pi, so those two will cancel each other out.

And the left-hand side, I've got 20 divided by pi, which means the diameter is equal to 6.

36619 and there's more decimals.

So, let's round it.

So, the diameter is equal to 6.

37 centimetres given to three significant figures.

Here's one for you to try.

The circumference of the circle is 125 millimetres.

Calculate the diameter.

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

Okay, once again, we're trying to find the diameter, so we'll use the formula, C equals pi d.

Substitute our numbers in.

We get 125 equals pi d.

Divide both sides by pi and we get d equals 39.

78873 and more decimals.

Round that and we get 39.

8 millimetres to three significant figures.

Now, the question didn't say what to round it to, so you might have rounded it differently, but it should be a similar number and hopefully you might have written what you rounded it to as well.

Here's another example.

The circumference of the circle is 18.

7 centimetres.

Calculate the radius this time.

So, because I'm calculating the radius.

I'm gonna use a formula, C equals 2pi r, 'cause that formula has the radius in it.

Let's substitute our number, 18.

7 in for C, and we get 18.

7 equals 2pi r.

And then, let's divide both sides now by 2pi, because I wanna get the r by itself.

If we divide the right-hand side by 2pi, then we've got 2pi divided by 2pi, which they'll cancel out and just leave the r there.

On the left-hand side, 18.

7 divided by 2pi, we get 2.

97619 and more decimals.

Let's round it to 2.

98 centimetres.

I've given that to three significant figures.

Here's one for you to try.

The circumference of the circle is 224 metres.

Calculate the radius.

Pause the video while you have a go and press play when you're ready to continue.

Okay, because we're calculating radius, we'll use a formula, C equals 2pi r.

Substitute r, 224 in.

Divide both sides by 2pi and we get 35.

65070, which needs to be rounded and we get 35.

7 centimetres if we give our answer to three significant figures.

Now, an alternative method for calculating the radius could be to first calculate the diameter, and then divide by two afterwards to get your radius.

I guess an advantage here could be that you don't have to worry about choosing the right formula in the first place if you just work out the diameter either time, either way.

And then, if you work out the diameter, you can then work out the radius once you know it.

For example, calculate the radius given your answers at two decimal places for this circle that has a circumference of 18 metres.

We could do it in one of two ways.

With method one, we could use the formula, C equals 2pi r, like we have been.

Substitute 18 in.

Divide both sides by 2pi and we'll get r as 2.

86478 and more decimals, which we round to 2.

86 metres.

Or method two could be to use the formula, C equals pi d.

Substitute our numbers in, divide by pi, and we get a diameter of 5.

729577 and more decimals, which rounds to 5.

73 metres.

And then, if that's the diameter, we need to divide it by two to get the radius, which gives us 2.

87 metres.

Now, hang on a minute.

In one method, we get 2.

86 metres and in the other method, we get 2.

87 metres.

Those answers are very close together, but they're not exactly the same.

Can you think why are these solutions just ever so slightly different? What's happened to cause that? Pause the video while you think about this and press play when you're ready to continue.

The problem here highlights a disadvantage with method two and that is what's happened is we've got our diameter of 5.

729577 cent, metres, sorry.

We've then rounded that, and then divided our rounded answer by two, which has given us our answer of 2.

87 metres.

Now, rounding reduces the accuracy of an answer.

And calculating with a rounded number is less accurate than calculating with the unrounded number.

So, really, this accuracy could be improved by having the diameter before we rounded it.

In other words, take out those last two lines.

Take the 5.

729577 and all the other decimals that were stored in that calculator and divide that by two, and we get a number of that round to 2.

86 metres.

Now, this problem could be avoided through method selection using the formula, C equals 2pi r, to calculate the radius, and using the formula, C equals pi d, to calculate the diameter.

Now, when the circumference of a circle is a multiple of pi, each side of the equation will have pi as a factor.

So, dividing pi by pi gives one, so that means the pi will cancel out on each side of the equation.

Here's an example.

The circumference of this circle is 18pi metres.

Find the diameter.

So, let's choose our formula, C equals pi d, because we're calculating diameter.

Substitute our circumference of 18pi in, so 18pi equals 18pi.

Now, when we divide both sides of this equation by pi, the pis on each side of the equation will cancel out, which means we've got 18 equals d.

Therefore, our circumference is 18pi metres and our diameter is 18 metres.

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

Which is the least accurate method for calculating the diameter of a circle? Is it, A, use the formula, C equals pi d, to calculate the diameter.

Is it, B, use the formula, C equals 2pi r, to calculate the radius, round your answer, and then multiply by two to get the diameter.

Or is it, C, use the formula, C equals 2pi r, to calculate the radius, multiply it by two to get the diameter, and then round your answer.

Pause the video while you think about this and press play when you're ready for an answer.

The answer is B.

The reason why that is least accurate is because rounding has taken place somewhere in the middle of the calculations They've calculated the radius, they've rounded it, and then multiplied the rounded answer by two to get the diameter.

Whereas in, C, rounding happens at the very end.

And also in, A, rounding happens at the very end as well, even though it's not said in there, but the rounding would take place at the end.

And here's another question to try.

The circumference of the circle is 14pi metres.

Calculate the diameter.

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

The answer is the diameter is equal to 14 metres.

That's 14pi divided by pi.

Okay, over to you now for task B.

This task contains two questions and here is question one.

Calculate the length labelled in each diagram and round any decimal answers to one decimal place.

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

Okay, and here is question two.

The diagram is a 2D representation of the International Space Station orbiting the Earth.

Now, the diagram's not drawn to scale.

Now, based on this, you've got three questions to answer A, B, and C.

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 work through question one.

In part A, we need to find the diameter, which is 4.

8 metres.

In part B, we need to find a diameter, which would be 15 metres.

In part C, we need to find a radius, which is 3.

4 metres.

Now, in part D, x is equal to the radius and y is equal to diameter.

So, it doesn't necessarily matter which one you work out first.

You could use one to work out the other.

So, you can work out x is 1.

4 metres and y is 2.

9 metres.

Now, notice that 2.

9 is not double 1.

4.

The reason why is because if you find x first, you need to double that before you round it to get the value of y.

Or if you find y first, you need to half it before you do any rounding to get the value of x.

And only round your answers at the end when you're writing them down as your final answers.

So, that's why we've got 1.

4 and 2.

9.

And then, part E, the length z goes from the bottom of the big circle all the way up to top of the little circle, and that is equal to the diameter of the big circle plus the diameter of the little circle.

Therefore the length of z is 15.

9 centimetres.

And in question two, so diagram represents a 2D representation of the International Space Station orbiting the Earth.

Now, in part A, it says the circumference of the Earth is approximately 40,075 kilometres.

It says, calculate the radius of the Earth to the nearest kilometre.

When we do that, we get 6,378 kilometres.

And part B, it says the circumference of the space station's orbit is approximately 42,590 kilometres.

It's bigger, because it's further out.

Calculate a distance between the space station and the centre of the Earth.

Well, that's the radius.

That's the radius of the space station's orbit.

So, if we do that, we get 6,778 kilometres.

Then, it says, part C, how far is the space station above the ground to the nearest kilometre? We can get that by taking our answer for part B and subtracting our answer for part A.

And preferably, doing that with the unrounded numbers to make it as accurate as possible.

Our answer would be 400 kilometres.

Absolutely fantastic work today.

Spot on.

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

The formula, C equals pi d, can be used to find a circumference of a circle when we know the diameter.

Or the formula, C equals 2pi r, can be used to find the circumference of a circle when we know the radius.

And both of those formulae can be used to find either the radius or the diameter when the circumference is known.

But answers that are written as decimals will need to be rounded, because it's impossible to write all those decimals out, because there's infinite many.

But when we round our answers to a super degree of accuracy, it does lose a bit of accuracy, which means the only way to write our answers exactly, in many cases with circles, is by writing it in terms of pi.

Well done, today.

Great work.