Lesson video

- Hello, I'm Mr. Coward and welcome to today's lesson on circumference of a circle.

For today's lesson, you'll need a pen and paper, a calculator, a ruler and a piece of string or a measuring tape.

If you could please take a moment to clear away any distractions and turn off any notifications, that would be great.

And if you can, try find a quiet space to work where you won't be disturbed.

Okay, when you're ready, let's begin.

Okay, so time for the try this task.

You'll need a measuring tape or a piece of string and a ruler.

Find circular objects in your house and measure the circumference and the diameter to the nearest millimetre.

So I've done this with three objects already.

I've measured the circumference, so all the way around the outside and I've measured the diameter going across from one edge to the other and I've recorded it to the nearest millimetre.

Now I want you, if you have these bits of equipment, if you have a ruler and a piece of string or a measuring tape to do this for two more objects.

Find the circumference and the diameter.

However, if you do not, do not worry, just find the ratio for these three.

So find two more if you can, and then find all the ratios.

How do we find the ratio? Well, we do the circumference divided by the diameter and that gives you the ratio.

Okay, so pause the video and have a go.

Pause in three, two, one.

Okay, welcome back.

Now, here are the ratios for the three that I had already and you can probably see that they're all kind of or they're very similar.

They're all three and a bit, and hopefully, the ones that you did as well were three and a bit.

Because if we were to measure them perfectly accurately and more accurately than we could ever do as humans with measuring tape and a piece of string, we'd actually get exactly the same number.

Exactly the same.

And this number is a really important number and we're gonna focus on this in today's lesson.

And we're gonna focus on this ratio between the circumference and the diameter.

So the diameter always fits into the circumference the same number of times.

So that's that special number.

So if I unravel my circle, well, how many diameters would fit in there? Three and a bit.

What is that bit? How many times bigger is the circumference than the diameter? Well, it's a very special number.

It's an irrational number.

It's a number that goes on forever and there's no pattern.

And that number is pi.

Some of you may have heard of this number before but if you haven't, don't worry.

So what is pi? Pi is a mathematical constant with the value of 3.

14159, and it goes on forever.

And there are no patterns in the digits.

So it doesn't repeat, it doesn't go 14591459.

It goes on like this.

And because there's so many numbers, infinitely many numbers with no pattern, you can actually find any single number in pi.

So you can actually find your phone number in pi and not in here 'cause here's only the first 1,000 digits, but maybe in the first million you might be able to find your your phone number.

Now, it's a very special number and you've just got to remember that pi is the ratio between the circumference of the circle and the diameter of the circle.

So the circumference of the circle is pi times bigger than the diameter.

So we get this formula, and let me write this down for you.

Circumference equals pi times diameter, which we sometimes write without the times sign.

We sometimes write that like that, pi d.

So the circumference is pi times bigger than the diameter and pi is this number here.

Now, you don't need to remember all that.

I like to remember these digits.

Some people just remember 3.

14 and 3.

14 is fine.

Some people know it up to a ridiculous amount of digits and they memorise it.

But unfortunately, I'm not one of those people.

I can only do about six.

Okay, so we are going to use the diameter now to find the circumference.

So do you remember the formula? Circumference equals pi times diameter.

So to find the circumference, we can do pi times eight.

I'll write my circumference there.

So we get eight pi, we like to write our number before we write the pi.

So it's just kind of the standard, like we like to write 3x like that with the number in front rather than X three.

Okay, so here is eight pi.

Now you can, on a calculator, you can work it out and you would get.

and that number goes on forever.

So well, which is better? Well, I think this is better.

This is more accurate than this can ever be.

So this is what most people prefer.

The more mathematicians prefer to write it in its exact form.

So you prefer to write it like eight pi.

However, sometimes you may be asked to write it in decimal form or round your answer to a number of decimal places.

So sometimes this form is useful and it can help give you a sense of the size of how big something is.

So both forms are useful, but mainly, in today's lesson, we'll be focusing on exact form.

So you can just get this from your calculator doing eight times pi but mainly, we're gonna be focusing on exact form.

So find the circumference of this circle.

Pause the video if you need to.

Pause in three, two, one.

Okay, welcome back.

Hopefully you did for the circumference, it's pi times four, which gives you four pi.

There we are.

Okay, what's changed now? Well, we don't have the diameter anymore, we have the radius.

So if the radius equals eight, what does the diameter equal? 16.

Double the radius.

So to find our circumference, we're going to do 16 times pi.

So circumference is 16 times pi, which equals 16 pi.

And I'm gonna put my units 'cause I actually forgot on the last side, which was kind of naughty of me.

So hopefully, you have learned from my mistake and will not forget units like I did.

Okay, so your turn, find the circumference of this circle.

Pause in three, two, one.

Okay, welcome back.

Hopefully, you've got that the diameter is eight.

So the circumference is eight times pi, which is eight pi.

And you remembered your units this time like I did.

Okay, a bit of a trickier question.

Show that the circumference is smaller than the perimeter.

So here I've got a square and the square's got area of 81.

So if the square has got area of 81, how long is this side? Well, it's the square root of 81.

The square root of 81 is nine, so that is nine centimetres.

So that means that my diameter is nine centimetres.

So the perimeter of my square is nine plus nine plus nine plus nine or nine times four centimetres.

And the perimeter of my circle, which is just a.

Circumference is just a fancy word for perimeter, is nine times pi, which equals nine pi.

But that's not that helpful here because it's quite hard to compare them.

So in this case, I would work out nine pi on my calculator, just doing nine times pi.

And that gives me an answer of 28.

3 to one decimal place.

Okay, obviously that would go on forever but I've just rounded it to one decimal place.

So now we can clearly see that my perimeter of my square is bigger than the perimeter of my circle.

Okay, this time, we're given the circumference, find the missing length.

Well, so do you remember how circumference equals pi times diameter? So that would mean that the circumference divided by pi is equal to our diameter.

So the circumference is 20 pi, sorry, is 20 pi.

So we divide that by pi, which gives us just 20.

So that means our diameter is 20 centimetres.

So we divide, when we've got the circumference, we divide by pi and that gives us our diameter.

So we do the opposite from going from diameter circumference.

So pause the video and have a go.

Pause in three, two, one.

Okay, welcome back and hopefully, you divided this number by pi.

So the diameter equals 12 pi divided by pi, which is just 12.

So well done if you got that.

And centimetres, which I almost forgot again.

Okay, so now it is time for the independent task.

I would like you to pause the video to complete your task and resume once you're finished.

Okay, welcome back.

Here are my answers.

You may need to pause the video to mark your work.

Okay, so now it's time for the explore task.

Each pattern is made out of congruent semicircle arcs.

Find the length of each pattern.

What do you notice? So pause the video to complete your task.

Resume once you're finished.

I do have a hint if you want it, but have a go first.

Okay, so here is my hint.

My hint is that imagine we had a full arc there, a full circle.

Well, that's half of a circle.

So what would the circumference be for the full circle? So what would it be for half a circle? Well, for the full circle, it would be 12 pi.

So for half the circle, it would be six pi.

We divide it by two.

And for this one, what's that length? Six.

It's half of that length.

What's that length? So we wanna work out that and then add that on and then add that on.

Okay, so now I've given you a little bit of a hint.

Pause the video and have a go.

Okay, so here are my answers.

Isn't that strange? Isn't that interesting? I mean, if you imagine walking that distance, you know, that distance looks like it's a much longer walk than walking this distance but it's the same, which is quite interesting.

Hmm, what do you think? Okay, so that is all for this lesson.

Thank you very much for all your hard work and I look forward to seeing you next time.

Thank you.