Lesson video

Hello, I'm Mr. Coward.

And welcome to today's lesson on area of circles.

For today's lesson, you'll need a pen and paper or something to write on and with and the calculator.

If you could take a moment to clear away any distractions, including turning off any notifications, that would be great.

And if you can try and 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.

Find the area of the square.

So this square and then find an area of this square.

Then find the area of this one and then this one, and then this one, and then finally estimate what you think that is.

So pause the video and have a go, pause in three, two, one.

Okay.

Welcome back.

Now hopefully you recognised that, that is not the full length of the square.

The full length is double that.

So the full length is two, cause that's one as well.

So we have a two by two square.

So two times two is four.

So that's got an area of four.

Well, this, everything's shaded, so that's an area four.

What about this? What fraction of the circle is shaded? A quarter.

So a quarter of four, one.

Okay what about this one? What fraction is shaded? What does it help to think that that's a quarter? So that's half of a quarter.

Half of a quarter, an eighth.

So what's half of one? Cause one is a quarter.

That is half of that.

So it's just a half.

Okay.

What about this one? It's trickier.

Isn't it? Well, you could split it up like this.

That's a half, that's a half.

That's a half.

That's a half.

So in total we have two, or you could think of it as that's half, that's half, that's half and that's half.

So it's exactly a half that's been shaded and half of four is two.

Okay.

So if that square is two, what's that? Well, hopefully you went for something around three and it's actually three and a bit and not just three and a bit, it's this.

Do you recognise that number? What is that number? Have we seen it before? The area of this of a circle radius one is PI.

Interesting.

Okay.

So I've got another try this task.

Now, what I want you to do is each square there is one centimetre squared.

So that length is one, two, three, four, five.

So we have a circle radius five, under 10 by 10 square.

Now what I want you to do, is I want you to estimate the area of that circle.

So pause the video and have a go.

Pause in three, two, one.

Okay.

Welcome back.

Now.

This is what I did.

So I did a rectangle.

I split into rectangles.

I've got one big rectangle there.

So that's a six by eight.

So that's 48.

Six, here we've got another six rectangle, a rectangle with area of six.

Now what where's this bit come from? Well, if you imagined me squeezing that bit in to there, along that bit there, you can see that that area is roughly the same as that area.

So I've said that if you put that on that side, you get a rectangle or a one by four rectangle.

So that gives me an area of four.

Do the same here.

Do the same here and again and again.

Okay.

Which squares I'm I missing? I'm missing this one, which I've said is about half.

This one, about half, this one, about half.

And this one, about half.

So that's two in total.

So I've got forty eight plus two.

Six twice or double six 12 and four times four.

If I add them together, I get an area of 78.

Okay.

So hopefully you got something similar than that.

Now is this estimate accurate? Well, when we'll calculate an area of circles, we can do more than just estimate.

There's actually a mathematical way of working out.

We've got a formula.

I'm going to tell you that the formula is this, PI r times r Okay.

I'm going to show you where that formula comes from in a minute with a demonstration, but let's just check if it's correct.

So it's got a radius of five.

So PI times five of.

Yep.

That's PI times five times five, which is 25 PI.

Now, if I type it into a calculator, 25 PI, I will get 78.

5 and that would go on forever.

Well, that's a pretty good estimation, isn't that?.

And in fact, if you keep going and you made it, you know, if you made that circle bigger, your estimate would just keep getting better and better and better.

Okay.

So now I've got a demonstration to show you where this formula comes from.

Okay.

So I have this demo that I want to show you.

Now I couldn't get my face on this demo, but I don't think that's important.

Now.

Watch what happens as I straighten out this conference.

Okay.

So what is that length there? Well, we could say that, that is PI times the diameter, but can you give me that one in terms of the radius? What would it be in terms of the radius? Well, the diameter is two times the radius.

So instead of saying that length is PI times diameter, we could say that length is two times the radius.

So that's the diameter times PI, like this two PI, okay, two times PI times the radius, the two times the radius that will give me the diameter.

So it's just the same.

It's just a different way of writing it.

So that length is two PI r Now I'm going to dissect this.

Okay.

I'm splitting up into lots of triangles.

Now, if I roll this out, can you see how all of those pieces, there are approximately triangles, on the total length for the base, is two PI r, and the total height is r.

So to find the area of these triangles, we would need to do a half times base two PI r times height.

So we do a half times two PI times r, and that would give me the half and the two would become one simplifies one.

So we'd get PI r times r.

And you can see this if I rearrange it like this, into what is kind of a parallelogram there.

So can you see that, the base is PI r because half of the circumference and the height is, r, so our area there is PI r times r.

And it becomes a little bit more obvious because you've got, you know, you've got little bits of the bottom there, which kind of get in the way but as you be covering more and more parts.

Can you see now that we've barely got anything coming out in the bottom and then I'm going to flip it over.

Okay.

That's that looks a lot like a parallelogram now.

And even if I increase it more and more and more and more, that becomes, well, it looks like a rectangle now.

And it looks like a rectangle with base PI r and height r.

So the area of that is PI r times r PI times radius times radius.

So that is our area of a circle, because if we undissect it, that's just what we started with.

And we just had that and we just had that and we moved it back into the circle.

So area is not changing.

And even if we have a lot of pieces, if we can bring that back into a circle.

Okay.

So that is the area of a circle, PI r times r.

Why? Because if we dissect and rearrange, it forms a parallelogram with base PI r and height r.

So now we're going to find the area of this circle.

Well, what's our formula? PI r times r.

Okay.

That's half the conference times by the radius.

So that would be like PI r times r.

That'd be like a parallelogram, even though it looks more like a rectangle.

Okay, so radius is eight.

So we've got PI eight or eight PI.

I'll write it as eight PI.

I would prefer to write eight times eight.

Eight times eight, 64.

So our area is just 64 PI.

And you could work that out as a decimal.

However, it's nice that this is more exact and we don't need to use a calculator for that.

So it's just a bit easier to leave it in terms of PI.

Okay.

And when we leave it in terms of PI, we call that exact fall.

So your turn, I would like you to have a go at this one.

So pause the video and have a go.

Pause in three, two, one.

Okay.

Welcome back.

Now, hopefully you did four PI times four, which gives you 16 PI and I forgot my units.

And it's sent to me as squared the units, because we're doing a length eight PI centimetres times eight centimetres, centimetres time centimetres sent me a squared.

Okay.

So we times two looks together.

So we get an area.

So we get squared units.

So really well done if you've got that correct, and really, really well done, if you remembered your units unlike me, cause I would have lost a mark for that.

Okay.

Find the area of the circle.

What's different now? Well, we don't have the radius.

What do we have? We have the diameter.

So what would the radius be? The radius would be four, half the diameter.

So using our formula, we've got four PI times four, which is the same one that you did before.

So we have 16 PI and our units, centimetres squared.

And we'll leave that in exact form.

Okay.

You have to find the area of this one.

So pause the video and have a go in three, two, one.

Okay.

Welcome back.

Hopefully you did.

You found that the radius was two.

So you did two PI times two and you got four PI centimetres squared.

Get real don't if you got that correct.

What fraction of the square is the circle? We have a trickier question this time.

The area of the square is 81.

So how long is that side? The square root of 81 is nine.

So the square root of 81.

That gives me nine.

Okay.

But that's not our radius, is it? That's our diameter.

So we still need to find the radius.

So the radius would be half of nine, which would be 4.

5.

Okay.

So our radius is 4.

5.

So we do 4.

5 PI times 4.

5.

And that gives us the area of a circle.

So 4.

5 PI times 4.

5, which is in terms of PI, it's 81 over four PI, as a fraction or as a decimal is 63.

617 et cetera.

Now I'm going to use the answer button on my calculator.

I'm going to use the answer button, divide that by the area of the square to find out what fraction of the square is the circle.

So my answer was 63.

61.

Now, the reason why I've used answer is just so that I don't get any kind of rounding error during my question.

So it's good to use the answer button.

So answer divided by 81.

And that gives me nought seven, eight, five et cetera.

Okay.

So what fraction is it? Well, that is approximately 785 over thousand.

And to be honest, I think it's asked for a fraction.

And so that's my fraction.

However, I think it's nicer probably as a percentage is question and I wish I'd actually made this a percentage question.

Okay.

So we have got here.

Do you recognise this number from before, 78.

5? Where did we see that? That's interesting, isn't it? Why, why is that true? Okay.

It's a little bit of a tangent, but I'm going to to tell you.

So imagine now we had a square and that, so we got a square here we are.

We've got circle of radius r.

Now the square, what would be the side the square? Side length would be two r.

Okay.

It'd be double the radius.

So the area of the square would be four r squared.

The area of the circle would be PI r times r.

Which we could think of as PI r squared times r squared.

Now, what we could do is we could simplify this r squared.

So we just get PI divided by four.

Now we've on a calculate you do PI divided by four.

You get this number there.

Okay.

So that's, it's actually always the same percent, always the same fraction of the circle, which is really interesting, I think.

Okay.

This time, given the area find the missing length.

So PI r times r equals 289 PI.

So what I'm going to do now, is I'm going to divide both sides by PI.

So I get r times r, should be PI there equals 289.

So how else could we write r times r? We could write that as r squared.

So r squared is 289, and then we do the square root and every square root we should get that square root of 289 is 17.

So that means r equals 17.

So the radius equals 17.

So that length there X equals 17.

So what did we do? We divided both sides by PI.

We wrote this r times r as r squared and then we square root at both sides.

The square root of r squared is just r.

On the square root of 289 is just 17.

So I'd like you to have a goal, try and find the radius for this question.

So pause the video and have a go in three, two, one.

Okay, welcome back.

Now.

Hopefully you've done this PI r times r equals 576.

So PI divided by PI divided by PI.

We get r times r equals.

Sorry.

PI r times r equals 576 PI.

We divide both sides by PI and then we get r times r equals 576.

Then we write that as r squared, we square root.

So we get r equals, what is the square root of 576 square root of 576 is 24.

So we get our radius is 24.

Really well done if you got that.

That was quite tricky.

Now know that my area has got PI in it.

If we didn't have a PI in it, we'd still divide by PI and we'd have to on a calculator, do that number divided by PI.

But because it has got PI in those PIs, just simplify to one.

So we just get that number.

So it's easier to work out the radius, when the area has got PI in it.

And I won't show you this lesson, but another lesson, I will show you how to work it out.

If it didn't have a PI in.

Okay.

So it's time for the independent task.

So 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.

Awesome.

Now it's time for the explore task.

So the face of this count is made partly of silver and gold coloured metal.

So we have the silver on the inside and then the gold on the outside.

What can you work out? So here we've got a few different statements of what people can work out.

Can you work these things out? So pause the video and have a go and resume once you've finished.

Pausing three, two, one.

Okay, welcome back.

Now, let's go for this.

I can work out the area of the face.

So what is the area of the total face? Well, that has got a radius three, so it's three PI times three, nine pie.

We can work that out.

I can work up the circumference of the face.

Well, for the circumference, we need the diameter.

The diameter is six because that's three and three.

So six, so six times PI, six PI.

And again, like we kind of mentioned when I was doing the demonstration and we can think of that as two times three times PI.

So that's where we get that six from, the two times three.

I can work up the air of the silver section.

What was the area of the silver section? Well, that's got radius two, so two PI times towo, four pie.

I can work out whether the gold or silver section is bigger.

How could you do that? How could you work out the gold area? Well, if the area of total thing is nine PI and the area of the silver is four PI.

The area of the gold section would be five PI.

Nine PI take four PI equals five PI.

So the gold section is bigger than the silver section.

Okay.

Is there anything else you could work out? Could you work out what fraction of the shape is gold? Could you work out what fraction of the shape is silver? What else could you find? Could you find the diameter of the silver section? Okay.

So there's a few things that you could work out.

So hopefully you had a goal and hopefully you worked loads of different things out.

And even if you just work these hours, still a really good effort.

So, well, don't on that is all for this lesson.

Thank you very much for your hard work.

And I look forward to seeing you next time.

Thank you.