Lesson video

Hello, my name is Dr.

Rowlandson, and I'm excited to be guiding you through today's lesson.

Let's get started.

Welcome to today's lesson from the unit of circle theorems. This lesson is called checking and securing understanding of the parts of a circle.

And by the end of today's lesson, we'll be able to identify the parts of a circle, including finding the perimeter and area of a sector.

This lesson will introduce a new keyword, which is subtend.

An angle can be subtended by a line segment or a curve.

And when that happens, the legs of the subtended angle meet the end points of the line segment or curve.

And we have a diagram on a screen to illustrate that.

We have an angle which is labelled a, and we can see that the legs of that angle meet two points, P and Q.

Those two points are the end points of the line segment PQ, which is the chord of the overall circle and also the end points of the arc PQ as well.

So we could say that the angle a is subtended by the chord PQ.

Or we could say that the angle a is subtended by the arc PQ.

Or we can turn the phrasing of that around and say that the arc PQ subtends the angle a.

Or we can say that the chord PQ subtends the angle a.

If you haven't got that now, don't worry about it.

We'll see plenty of examples of the usage of this during today's lesson.

And here are some previous keywords that will be useful during today's lesson as well.

Feel free to pause the video if you want to remind yourself what any of these words mean, and then press play when you're ready to continue.

The lesson is broken into two learning cycles, and we're going to start by using the parts of a circle.

Here we have Andeep and Aisha, and they are thinking about words which are to do with the parts of a circle.

And Andeep is a little bit distressed by this.

He says, "There are so many different parts of a circle that I have to remember." Aisha says, "Yeah, I know.

How many words relate to circles do you know though?" I wonder how many words relate to circles you know.

Pause the video while you think about as many as you can.

Perhaps write down as many words as you can think of to do with parts of a circle and then press play when you're ready to continue.

Now I wonder how many words you thought of there.

It may be that your words are categorised in some way.

or it may be that they're just a big jumble on your page.

Let's think about how we could sort some of these words which have to do with parts of a circle out into a bit more of a structured way.

Some of those words relate to lengths, and some of the words relate to areas.

And of the words that relate to lengths, some of them are to do with lengths that are inside the circle, and some are to do with lengths that are either on the circle or outside the circle.

Let's take a look at some examples.

For lengths that are inside the circle, we have radius.

We have diameter.

And also we have chord.

For lengths which are either on or outside the circle, we have circumference, we have arc, and we have tangent.

And then for parts of a circle which are areas, we have sector, and we have segment.

Now we've organised those words.

Andeep says, "That's great, but what is the difference between all these different parts though?" Let's take a look at some together.

Let's start by looking at the difference between a chord and a segment.

A chord is a length that is inside the circle, and a segment is an area.

So let's take a look at what they look like.

Here in the screen now we have a chord.

It is a line segment that goes from one part of the circle to another part of the circle.

A chord is any line segment whose two endpoints touch the circumference of the circle.

And a chord does not need to pass through the centre of the circle.

If it does, it's a diameter.

Here we have a segment.

A segment is an area, and segments are created by dividing a circle into two parts using a chord.

So a chord is the length, and it cuts a circle into two parts.

Each part's an area, and they are segments.

The segment which is highlighted here is under 50% of the area of the full circle.

So these segments are called minor segments, whereas this segment now which is highlighted, that is over 50% of the area of the full circle, so it's a major segment.

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

Here we have four diagrams labelled one to four and four names or descriptions labelled a to d.

Match the part of the circle with its name or description.

Pause the video while you do that and press play when you're ready for answers.

Let's see how we got on.

Here are the answers.

The diagram in one is a semicircle.

It's a special type of segment that has exactly 50% of the area of the circle.

And diagram two is a major segment.

It is an area of the circle that is constructed from the chord and an arc, and it is over 50%.

The diagram in three is not a segment.

That is not constructed from a chord.

And four is a minor segment.

That is because we have an arc and a chord, and it makes an area which is less than 50% of the circle.

Let's take a look at some more words now, in particular, radius, arc, and sector.

An arc is a length that is on the circle, and it looks something a bit like this.

It can be smaller than that, or it can be bigger, but an arc is part of the circumference of a circle.

Now if we draw a radius from the centre of the circle to each end point of that arc, we would have a sector.

A sector is formed by two radii and an arc that is joined together.

And a sector is an area.

It's the area enclosed between the radius, radius, arc.

It's this area here.

Now a minor sector is under 50% of the full area of the circle, and a major sector is above 50% of the area of the full circle.

So let's check what we've learned with that.

Here again, we have four diagrams and four descriptions.

Match the parts of the circle with its name or description.

Pause the video while you do that and press play when you're ready for answers.

Let's see how we got on.

Here are the answers.

The diagram in one, again, is a semicircle.

Now, earlier we said that a semicircle is a special type of segment that is 50% of the overall area of the circle.

Well, it's also a special type of sector where the area is 50% of the circle as well.

It's a special type of sector that is made from a diameter composed of two radii.

Now two is not a sector.

That's because those two line segments are not radii of the circle.

They don't go from the centre.

Now diagrams three and four are both sectors because those line segments are radii, and we can see that because they go from the centre of the circle.

But three is a minor sector because its area is less than 50% of the area of the full circle.

And four is a major sector because its area is greater than 50% of the full circle.

So let's look at some more words.

We have yet to talk about circumference or tangent.

Circumference is a length that is on the circle.

In particular, it is the distance around the circle, or the perimeter of the circle.

A tangent is something a little bit different.

A tangent is a line or a ray or a line segment that intersects a circle exactly once on its circumference, like we can see in this diagram on the screen here.

We can see it in this case it's a line segment, and it intersects at a single point which is marked on the diagram.

Now this line segment is not a tangent.

Yes, it does only intersect the circle once, but if we extend the line segment, then it would intersect the circle a second time.

So we should always think about that, what would happen if you extended the line segment.

And if you did, if you turned it into a line with infinite length, it should only intersect the circle exactly once.

So let's check what we've learned with that.

Here we have six diagrams. They all have a circle and some kind line segment.

What I want to know is which of those line segments are tangents to the circle.

Pause the video while you write down your answers and press play when you're ready to see what the answers are.

The answers are three and five.

In these cases, we have a line segment, which, even if it was extended, it would only intersect the circle once.

Here's another question.

We have four diagrams and four descriptions or parts of a circle.

Match the parts of a circle with its name or description.

Pause the video while you do that and press play when you're ready for answers.

Here are our answers.

One is a chord.

It's a line segment that goes from one part of the circle to another.

Two is a radius.

It's a line segment that goes from the centre of the circle to its circumference.

Three is none of the above.

Four is a tangent because it's a line segment that touched the circle exactly once and would still only touch the circle once even if you extended it.

So now we've recapped the names of the parts of a circle.

Let's consider how they may relate to angles.

Angles can be created from different lengths on a circle, and we say that these angles have been subtended.

Let's take a look at some examples.

Here we have a chord AB on that circle.

We also have a point Y which is on the circumference of the circle.

Let's create an angle which we'll call alpha.

The angle alpha degrees can be subtended at point Y by the chord AB by drawing the two legs of the angle like so, from A to Y, from B to Y.

So now we have an angle alpha degrees which is subtended by the chord AB.

Here's another example.

We have the angle alpha degrees, which is defined by the legs AY and BY, and the arc AB subtends the angle alpha.

And here's another example.

Here's an arc AB and a point O, which is the centre of the circle.

If we create an angle at O and call it beta, the angle beta degrees can be subtended at O from the arc AB.

It would look something a bit like this.

And then here's another example.

Here we have a radius AO and a point P on the outside of the circle, and we can say angle gamma degrees can be subtended at point P from radius AO.

And it'll look something a bit like this.

So let's check what we've learnt.

Here we have a diagram with points U, T, and Y, and let's now create an angle, and we'll call the angle alpha degrees.

Which of these sentences are correct for this diagram? Pause the video while you choose from one of them and press play when you're ready for an answer.

The answer is c.

Angle alpha degrees has been subtended by the tangent UT.

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

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

Pause the video while you do this and press play when you're ready for more questions.

And here are questions two and three.

Pause the video while you do these and press play when you're ready for more questions.

And here is question four.

Pause the video while you do this and press play when you're ready for question five.

Here is question five.

Pause the video while you do this and press play when you're ready for answers.

Okay, let's take a look at some answers.

In question one, you had to join different combinations of two or more of these circle parts to make as many different other circle parts as you could.

Now there are lots of different combinations you can use of this, so let's just take a look at a few examples.

Here's one example.

You could join the arc from a with the arc from b with the radius from c and the radius from e, And they would make a semicircle.

Or you could join the arc from a with the radius from c with a radius from d, and that would make a minor sector.

Or you could join the chord from g with the arc from i, and that would make a major segment.

And then in question two, you had two circles.

Each circle had the same two points on, Y and Z, to create that chord YZ.

And you had to subtend an angle at point A and point B on those circles.

When you do that, it should look like the diagrams on the screen here, and both of those angles should be equal at approximately 55 degrees.

In question three, you had a circle, and you had an arc, and you had to subtend an angle at point O from the arc VW.

And what's the name on that shape which is bound by the arc and also the two legs of its angle? It's a minor sector, or you can just have sector.

In question four, you had a circle with a radius and a tangent drawn on, and you had to draw a line segment PK, which you can now see on the screen.

And you have to complete the sentence.

Well, the line segment PK is a tangent of the circle because it only intersects the circle once.

That's at point K.

And then when you have to measure the length of PK and PT, what you should have noticed is both line segments are the same length.

And then in question five, you had a circle with a chord EF and two points, G and H, which are on the circumference.

In a, you had to subtend an angle at point G from the chord EF and measure it.

It should have been 114 degrees or about that.

And then for b, you had to subtend an angle at point H from the chord and measure that.

That should be about 66 degrees.

And then for c, you had to complete the sentence "Angle EGF and EHF lie in different segments of the circle," and then write down an observation about the angles EGF and EHF.

Well, they both sum to 180 degrees.

Well done so far.

Now let's move on to the second part of this lesson, which is looking at calculations with parts of a circle.

Here we have four diagrams containing circles and another part of a circle.

A has a radius.

B has a tangent.

C has a diameter.

And d has a chord.

And Andeep says, "Which of these circles can I easily find a circumference of, and how do I know?" Maybe think about this yourself.

Pause the video while you think about which of those circles could you easily find the circumference of based on the information given.

Then press play when you're ready to continue.

Well, you could easily find a circumference for circles a and c.

That's because a has a radius, and c has a diameter.

We can easily find the circumference of a circle from its radius or diameter.

Tangents and other chords can't easily be used to calculate a circumference.

The formula for the circumference of a circle is C equals two times pi times r, where C stands for circumference, and r is the radius of the circle.

That's gonna also be rewritten as C equals pi times d, where d is the diameter of the circle because don't forget the diameter is double the length of a radius.

Let's check what we've learnt.

You've got four circles labelled a to d, and you've got four answers labelled e to h.

Match the circle to its circumference.

Pause the video while you do that and press play when you're ready for answers.

Let's take a look at some answers.

A, circumference cannot be found, For b, circumference is 14 pi centimetres.

For c, the circumference is 22.

0 centimetres.

And for d, circumference is 87.

96 centimetres.

So we've talked about how to find the circumference of a circle based on its parts.

You'd need to find the radius or diameter.

Well, Aisha says, "It's similar for the area of a circle.

I need to look out for the radius, or half its diameter." For example, here we have quite a complex diagram, and if we wanted to find the area of that circle based on the information given, well, we wouldn't use the length of the chord, and we wouldn't use the length of that tangent we can see there, but we would use the length of the radius.

That's because the formula for the area of a circle is A equals pi times r squared, where r stands for the radius.

Well, in this case, the radius is 4.

8 centimetres, so we can use that to find the area of the circle.

The length of the chord and the tangent, they can't be used to find the area of the circle.

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

Match each circle to its area.

Pause the video while you do that and press play when you're ready for answers.

Okay, let's take a look at some answers.

The area of a is 113 centimetres squared.

The area of b is 254.

47 centimetres squared.

The area for c, that can't be found.

That's because we are given a chord, We're given a tangent.

But we don't have either the radius or diameter.

And for d, well, the area of that one is 144 pi centimetres squared.

So we've recapped how to find the area or circumference for a full circle.

But what about parts of a circle? We can also find the area and perimeter of a circular sector by considering the angle at the centre of the circle that's subtended by the arc.

For example, here we have a circle with points A and B on, and they make the arc AB, and we have also a point at the centre, O.

If we subtend an angle at a point like this, we have a sector.

Now when it comes to finding the area of the circular sector, we could do pi times r squared, which is the area of the full circle.

But we don't have a full circle here.

We have a fraction of the circle, and that fraction is theta, which is the angle at the centre of the circle over 360, which is the angle around a point.

So if we do that fraction multiplied by pi multiplied by r squared, we get the area of the sector, where r is the radius, and theta is the angle subtended by the arc AB.

For example, if the radius was 48 centimetres, and the angle subtended by the arc was 125 degrees, then the area would be 125/360 multiplied by pi multiplied by 48 squared.

And that would give 800 pi, or we could write it as 2,510 centimetres squared if we round our answer to three significant figures.

So what about perimeter? Well, to find the perimeter of a circular sector, we can consider the three lengths, the length of the two radii and also the length of the arc AB.

In this particular case, we're thinking about these three lengths here.

Now we know two of the lengths.

OA is 48 centimetres, and so also is OB because that's a radius as well.

But we don't yet know the length of the arc AB.

Well, to find the length of this circular arc, we need to do two times pi times r because that's how you find the circumference of a full circle.

But we don't have a full circle here.

We have a fraction of a circle.

So we'd need to multiply by theta over 360, where theta is the angle at the centre of the circle.

So in this case, we'd do 125 over 360 multiplied by the circumference, two times pi times 48.

And that would give 100 over three pi, which you could also write as 105 centimetres if we round our answer to three significant figures.

Now we haven't got our final answer here for the perimeter of a sector.

That number is just the length of the arc.

So to find the perimeter of the circular sector, we need to add that to the other two lengths that make up the sector.

We'd do 100 over three pi plus 48 plus another 48, and that would give this answer here, which we could write as 201 centimetres if we round our answer to three significant figures.

Okay, let's check what we've learnt.

Here we have a diagram where we have an angle which is subtended by arc TU at point O, and you've got some properties of the sector.

We have the area, the arc length, and the perimeter.

And you've got some values from d to h.

Match the properties of the sector to the values.

And in some cases, you may be matching more than one thing to another.

Pause the video while you do this and press play when you're ready for answers.

Here are your answers.

The area could be written as 4,640 pi, or it could be written as 14,600 to three significant figures.

The arc length is 116 pi, and the perimeter is 524.

42.

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

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

Pause the video while you do this and press play when you're ready for more questions.

And here are questions three and four.

Pause the video while you do this and press play when you're ready for answers.

Okay, let's see how we got on.

In question one, you had to calculate the area and circumference of the circle in the diagram, and you were given more information than you needed.

Your job was to select the correct information and the correct formulae in order to perform the calculations you needed.

If you did that, the area will be 10,404 pi units squared, and the circumference will be 204 pi units.

In question two, you had to start by subtending an angle from arc YZ at point 0 and measure it.

And if you did that, you should have got something around 80 degrees.

And then you needed to use that to find the area of the sector.

If you did that, you should get 2,040 units squared if you measured accurately as 80 degrees.

If you measured it slightly inaccurately in part a, your answer will be a little bit different to that but not too different.

In question three, you had to use information given to sketch the circular sector VOW.

And if you did that, it may look like the one on the screen, or it may be in a different orientation.

And then you had to also state the size of the angle and also the radius.

Well, the angle is 144, and the radius is 24 centimetres.

And then use that to calculate the perimeter of the sector.

Well, you should get 19.

2 pi plus 48 centimetres.

And then with question four in part a, you have to subtend an angle at point O from the code PQ and measure it.

If you did that, you should get 90 degrees.

In part b, by first measuring the lengths PM and MQ, you had to find the area of the triangle PMQ.

Now, checking this answer can be a little bit tricky 'cause it depends on how this question was presented, whether it's on a screen, or whether it's printed.

And the settings of those can affect the size of the circle.

The lengths that you measure therefore for PM and MQ may vary depending on how large the diagram is for you.

However, no matter how large the diagram is for you, MQ should be double length of PM.

Therefore, the area of triangle PMQ should be whatever PM is squared, centimetres squared.

Fantastic work today.

Now let's summarise what we've learnt.

A circle may have different parts, may have a radius, diameter, a chord, a circumference, an arc, a tangent, a sector, and a segment.

An angle can be subtended by the end points of any line segment or curve.

And this means an angle is created at a specific point by drawing the two legs of that angle from the points of the two endpoints of the line segment or curve.

A sector can be formed by subtending an angle from the endpoints of a circular arc.

And the area, arc length, or perimeter of that sector can be calculated by measuring or calculating the subtended angle.

Well done today.

Have a great day.