Lesson video

Hello, my name is Dr.

Rowlandson, and I'll be guiding you through today's lesson.

Let's get started.

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

This lesson is called using the formula for the area of a circle.

And by end of today's lesson, we'll be able to use that formula to calculate either the area or a missing measurement on a circle.

Here are some keywords from previous lessons that we'll be using again in today's lesson.

So you may wanna pause the video while you refamiliarize yourself with these words and then press Play when you're ready to continue.

This lesson has two learning cycles.

In the first learning cycle, we're gonna be finding areas of circles by using the formula.

Then the second learning cycle, we'll be finding either the radius or the diameter when the area is given to us.

Let's start off with finding areas of circles by using the formula.

The formula A equals pi r squared can be used to calculate the area of a circle, and A stands for area and r represents the radius.

Now, calculations can be performed accurately with pi on a scientific calculator by pressing the Shift key and then pressing the pi button.

Now on this calculator, pi is above seven, so I'd use that button, but on other calculators, it might be in a different place.

Answers can be given in terms of pi or they can be converted to a decimal by, in this calculator, pressing the Format button and selecting decimal.

The area of a circle can be calculated when the radius is known by using the formula A equals pi r squared, where r is the radius.

So for example, here we have a circle with a radius of 10 millimetres.

If I wanna find the area of it, I would go area equals pi times 10 squared, which calculates to get 100 pi millimetres squared.

Now, areas can be written accurately in terms of pi, or they can be written as decimals, but when we write 'em as decimals, those decimals may go on infinitely.

So we may need to round our answers to an appropriate degree of accuracy.

Now, when the diameter is given, the radius can be obtained by halving the diameter.

We may wanna do this because the formula A equals pi r squared uses the radius, not the diameter.

Once we've done that, we can use that formula to calculate the area.

So for example, here we have a circle with a diameter of 20 millimetres.

If a diameter is 20 millimetres, the radius must be 20 divided by two, 10 millimetres.

Now we know the radius is 10 millimetres, we can calculate the area.

We can do area equals pi times 10 squared and do all the same things we did last time to get 314 millimetres squared.

So you may need to determine whether a length that is given to you is equal to the radius or the diameter before you can calculate the area.

Don't just substitute whatever number you are given into pi times r squared 'cause it might not be the radius.

For example, here we have a circle and we can see that the distance from the centre of the circle to its edge is eight centimetres.

So that means that must be the radius.

Therefore, we can do pi times eight squared to get the area of 64 pi centimetres squared.

But in this example, the distance from one side of the circle to the other, the furthest distance is eight centimetres.

That means that must be the diameter.

So the diameter's eight centimetres.

We need to get the radius first.

The radius is four centimetres, and then we can find the area.

Area equals pi times four squared, just 16 pi centimetres squared.

Always double check, have you got the radius or the diameter? Let's check what we've learned there.

Here we have a circle and a measurement of 12 metres is given to you.

Which calculation would find the area of this circle? Your 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.

That 12 is a radius, so we can substitute it into our formula, pi times r squared.

So the calculation pi times 12 squared would give us the area.

Which calculation would find the area of a circle that has a diameter of eight? Options are A, B, and C.

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

The answer is A.

If the diameter is eight centimetres, then the radius must be four centimetres, and then we substitute that into our formula to get pi times four squared.

Jacob has calculated the area of the circle and given his answer in terms of pi.

Write Jacob's answer as a decimal, rounded to three significant figures.

So there you can see his calculations so far.

Pause the video while you finish this off, and press Play when you're ready for an answer.

The answer is 50.

3 centimetres squared.

In some cases, you may need to first determine the length of the radius or the diameter before calculating the area because it might not be given to you quite so clearly.

For example, here we have a circle and there are no numbers given to us, but it is overlaid over a grid.

So we can see here that the diameter, which is the distance from one side of the circle through the centre to the other side, is equal to eight units.

You could even look at it and think, well, the distance from the centre to the outside is four units.

Either way, we've got our radius now of four units.

So the area is pi times four squared, 16 pi unit squared or 50.

3 units squared.

Here's another example of this.

We have a circle that's on a coordinate grid, but we're not given any of the lengths, either the diameter or the radius, but what we are given are two coordinates.

The coordinate five, five is in the centre of the circle and the coordinate nine, five is on the edge of the circle.

So how far is it from the centre to the edge of the circle? Well, both these coordinates have a Y ordinate of five.

So the height doesn't change between those two points, but the X ordinate of the centre is five, and the X ordinate of that point in the circumference is nine, which means it must be four units across.

So now we've got our radius of four, we can calculate the area.

Substitute it into our formula, and we get 16 pi unit squared or 50.

3 units squared.

Let's check what we've learned with that.

We've got a circle again overlaid by a grid, which of these calculations would find the area of the circle? Your options are A, B, or C.

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

The answer is A.

In this circle, the distance from the centre to the edge is two, so that's our radius.

You might need to get that by getting the distance of the diameter first, which is four, and halving it.

But once you've got that radius of two, substitute it into our formula, pi times two squared.

Which calculation, this time, would find the area of this circle? You've got a circle and axes.

You've got a coordinate at the centre and a coordinate at the circumference.

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

The answer this time is B, pi times five squared.

The distance between those two coordinates, well, the X ordinate is the same in each, so those must be directly above each other.

And the difference in height between those two coordinates is five.

So the radius must be five, therefore the area's pi times five squared.

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

This task contains two questions and here is question one.

You need to find the area of each circle, giving your answer as a decimal rounded to three significant figures.

Now, in each case, the diagrams are not drawn to scale.

So do think about the information that you are given in those diagrams. Pause the video while you have a go at this and press Play when you're ready for question two.

And here is question two.

We've got five parts to this question.

Each part has a circle in it somewhere.

You need to use the information that you are given to find the areas of those circles, giving your answer this time in terms of pi.

In each case, you need to figure out what the radius is before you can get going with the question properly.

So make that your priority.

Pause the video while you have a go at this and press Play when you're ready to go through some answers.

Okay, well done with that.

Here's question one.

Part A, a circle of a radius of six metres would've an area of 113 metres squared, rounded to three significant figures.

In part B, a circle of a diameter of six metres would have an area of 28.

3 metres squared 'cause the radius would be three.

In part C, that measurement given is the radius.

So do pi times four squared to get 50.

3 centimetre squared.

In part D, that measurement is the diameter, so we need to half it first to get 11 millimetres and then use that to get an area of 380 millimetres squared.

In part E, that measurement is equal to the radius.

So we'll get an area of 121 centimetres squared.

In part F, that 8.

4 is equal to the diameter.

So half it first and then get your area of 55.

4 centimetres squared.

In part G, that nine is equal to the diameter; it's from one side of the circle to the other.

Half it first, and then you've got your area of 63.

6 centimetres squared.

And then finally part H, well, you can use either of those.

The 28 is the diameter, so you could half that to get the radius.

Or you can see how 14 is just a radius 'cause it goes from the centre to the top of the circle.

Either way, you'll be using 14 in your formula, and you'll get an area of 616 millimetres squared.

And let's now work through question two.

Part A, the distance from the centre of the circle to the circumference is 12 centimetres.

It doesn't explicitly say radius, but that's what the radius means.

So we'll use the 12 centimetres in our formula and get an area of 144 pi centimetre squared.

In part B, it says the furthest distance from one side of the circle to the other is 12 centimetres, so that must be the diameter.

So we need to half that to get six and then put that into our formula to get 36 pi centimetres squared.

And part C, we've got a grid over our circle.

We can see at the distance from the centre of the circle, which has a spot on it to the edge is three units.

Substitute that into our formula to get nine pi units squared.

And part D, we've got a coordinate at one edge of the circle and another at the other edge of the circle.

And we can see there's a line segment joining those two, which goes to the centre.

That means that line segment, that dash line segment must be diameter.

What's the distance of that? Well, it's between four and 14, so it's a diameter of 10.

That means there's a radius of five.

So our area is 25 pi unit squared.

And then part E, we've got our circle over an isometric grid, lots of triangles on a grid there.

We need to figure out the radius.

We can either do that by going from the centre upwards or downwards or maybe diameter to the right or diameter to the left.

Either way, we can see that the radius is equal to two units.

So if we substitute that into our formula, we get four pi units squared.

Great work so far.

Let's now move on to the second part of today's lesson, which is finding the radius or the diameter when the area of a circle is known.

In cases where the area is known, the formula A equals pi r squared can be used to calculate the radius.

This can be done by substituting the known value into the formula to form an equation, and then rearranging the equation to find your radius.

And this will usually involve dividing the area by pi and then finding the square root.

And this is gonna give us decimals and we need to be really careful with our decimals 'cause to maintain accuracy, do not round any decimals until the final answer has been obtained.

If you round decimals midway through your calculations and then perform calculations on those rounded numbers, our answers will be inaccurate at the end.

The more we round stuff midway through, the more inaccurate our answer becomes.

So save any rounding until the very end.

So here's an example which you can work through together now, and you'll get one very similar in a few minutes time.

The area of the circle is 200 centimetres squared.

Calculate the radius, giving your answer to two decimal places.

Let's do that together now.

Let's take our formula, A equals pi r squared.

And we know the area.

So let's substitute 200 in for A to get 200 equals pi r squared.

Now we need to rearrange this to get the value of r.

So let's first do that by dividing both sides of our equation by pi.

And that'll give us 63.

6619 and more decimals equals r squared.

Now I'm not going to round that decimal just yet.

Every time I do a calculation of it, I'm gonna use as much that decimal as possible or by using the answer key on my calculator to use the entire decimal.

So we need to now square root both sides of this equation and that will give us r equals 7.

9788 and more decimals.

And now we're ready to round our number to get a radius of 7.

98 centimetres given to two decimal places.

Okay, it's gonna be over to you in a second to do the same thing yourself.

However, you might be thinking at this point, oh, there's a lot of calculations, there's a lot of steps to this.

Try and do all that in one go might be a bit tricky.

So I'm gonna give you a series of three questions.

In the first question, I'll give you a lot of the calculations already done, and then you just need to finish it off.

And the second question, I'll give you a fewer calculations so you have to do more work.

And the third question, you can do it all yourself.

So here's your first question.

The area of the circle is 150 centimetres squared.

The first four lines of calculations are provided here for you.

And it says finish calculating the radius, giving your answer to two decimal places.

So you just need to finish off the last few steps of working in this question.

Pause the video while you do that and press Play when you're ready to continue.

Okay, so the next step we need to do here is to square root both sides.

And that will give us 6.

9098 and some more decimals equal to r.

The last thing we need to do is round our answer to two decimal places, and that gives us r equals 6.

91 centimetres.

So here's another one this time.

The area of the circle is 120 centimetres squared.

And I've only given you the first two lines of working this time.

You need to finish calculating the radius and again, give your answer to two decimal places.

Pause the video while you do this and press Play when you're ready to continue.

Okay, so our next line of working is we need to divide both sides by pi.

And once we've done that, we'll get 38.

1971 and some decimals equals r squared.

Do not round those decimals until the very end.

We then need to square root both sides to get 6.

1803 and more decimals equals r.

And then finally, we're ready to round our answer to 6.

18 centimetres.

And here's one more, and you need to do it all yourself this time.

The area of the circle is 80 centimetres squared.

Calculate the radius, giving your answer to two decimal places.

Pause the video while you do this and press Play when you're ready to go through it together.

Okay, so first, let's write down our formula, A equals pi r squared.

Let's substitute 80 in for A, divide both sides by pi.

Don't round our answer at this point 'cause we've got a square root r answer, and then we will then round it to get 5.

05 centimetres.

Well done with that.

Now, once the radius has been obtained, it can be used to calculate the diameter because the diameter is twice the length of the radius.

So for example, here we have a circle with an area of 200 centimetres squared.

We could calculate the radius by doing the same things we've done so far, 200 equals pi r squared, divide both sides by pi, get our answer and square root it.

And then we've got our radius.

Now this is the radius, but we're trying to get the diameter, so we're not gonna round this number at this point because we do not round any numbers before the final solution.

We want to calculate the diameter now, so we'll use our radius, we'll times it by two to get our diameter.

And then finally we'll round our answer to two decimal places, for example, which is 15.

96 centimetres.

Alternatively, once the radius has been obtained, we could also use it to calculate the circumference by substituting it into the formula, C equals two pi r.

For example, we've got this circle again, we could calculate the radius in the same way.

And then take that unrounded number for the radius and substitute it into our formula for circumference to get two times pi times 7.

9788 and all the other decimals, preferably using the answer button on your calculator.

And you'll get C equals 50.

1325.

And we can round that to get our circumference of 50.

13 centimetres.

Let's check what we've learned there.

Sofia is using the area of a circle to calculate its radius.

Now she's got the first three lines of her work in there.

What calculation should she perform next? You got four options, A, B, C, and D.

Pause the video, choose one, and press Play when you're ready for an answer.

The answer is D.

Her next step is to square root both sides of that equation.

So she should be square rooting the 95.

49 and all of the decimals.

Sofia is using the area of a circle to calculate its diameter.

What calculation should she perform next at this point? You've got options A, B, and C.

Pause the video while you choose one and then press Play when you're ready for an answer.

The answer is A, she's gonna want to take her radius of 9.

7720 before it's rounded and times it by two to get the diameter.

Because the diameter is twice the radius.

Sofia is using the area of a circle to calculate its circumference.

What calculation should she perform next? Pause the video while you think about this.

Choose either A, B, C or D and then press Play when you're ready to continue.

The answer is D.

She's got the radius to get the circumference, she needs to substitute it into the formula, two pi r.

So she'll be doing 9.

7720 and all the other decimals times two times pi, and that will give a circumference.

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

This task contains one question, and here it is.

The area of the circle is 500 centimetres squared, and you need to calculate the radius, the diameter, and the circumference of the circle, giving each of your answers to one decimal place.

And do be careful that you do not use rounded numbers from one part of a question to then calculate the answer to another part of the question.

Try and use the unrounded numbers for each question.

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

Okay, well done with that.

Let's go through some answers.

Part A, calculate the radius, that is 12.

6 centimetres.

Part B, calculate the diameter, that's 25.

2 centimetres.

And in part C, the circumference is 79.

3 centimetres.

Fantastic work today.

Absolutely spot on.

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

The area of any circle can be calculated when the radius is known, and that is by substituting the radius into the formula, area equals pi times radius squared or A equals pi r squared.

If the radius is not known but the diameter is known, that's okay as well.

'Cause the radius can be found by halving the diameter, and if the area of the circle is known, well, we can use that to calculate either the radius or the diameter by substituting the area into the formula, A equals pi r squared, and then rearranging the equation to get the radius.

And that can be used to get the diameter, could even be used to find the circumference.

Any answers that are decimals will need to be rounded to an appropriate degree of accuracy.

But do bear in mind that an answer can be written exactly by writing it in terms of pi.

Thank you.