Lesson video

Hello everyone and welcome to today's lesson on 2D shapes.

Thank you all for joining me, Mr. Gratton, where today we will: identify and name a range of different parts of a circle, and use appropriate formulae to find the perimeter and area of a circle.

Pause here to check through some keywords about circles that we will be using today.

And pause here to look at two new circle keywords, the chord and the segment, which we will look at in more detail later on.

First up, let's look at some features of a circle that you may be familiar with, as well as potentially some new ones.

Speaking of familiar parts of a circle, pause here to think about or discuss whether you can name these three parts on screen.

Andeep remembers that they are called the circumference, radius, and diameter.

Let's have a look at them in a little bit more detail.

The circumference of a circle is the perimeter of the circle.

For example, the circumference of this circle is its full perimeter.

But it is not, for example, only part of the perimeter or part of the perimeter joined up with a straight line segment.

The radius of a circle is a line segment that starts from the centre of the circle and ends at its circumference.

For example, if for these two circles, we have two centres, then these are two radii or radiuses, as one endpoint is at the centre of the circle and the other endpoint is at the circumference of that circle.

However, these are not radii.

The first, because the endpoint of the line segment does not touch the circumference, and the second, because the other endpoint of the line segment does not touch the centre of the circle.

The diameter of a circle is any line segment that both starts and ends on the circumference of the circle, but also passes through the centre of the circle.

For example, for these two circles, each with a centre labelled, then these are both diameters as they pass through the centre of the circle with both their endpoints touching the circumference of the circle as well.

However, these are not diameters.

The first, because only one endpoint of the line segment touches the circumference, and the second, because the line segment does not pass through the centre of the circle.

For any circle that you draw or are given, its diameter is always exactly twice the length of its radius.

For example, the radius of this circle is 12 centimetres long.

Any diameter of the same circle or of a congruent circle will be 24 centimetres.

Double 12.

This distance on this circle is 23 centimetres, so it is either a line segment that does not pass through the centre of a congruent or non-congruent circle or a line segment that is actually a diameter but of a different circle, different to the one that we see above.

For this check, pause here to match all three parts of the circle with its name and identify the fourth circle with a part that doesn't have a name.

Here are your answers.

Circle 3 is not a radius as the endpoint of the line segment is not at the centre of the circle, but beyond it.

However, it is also not a diameter as that same endpoint isn't quite at the circumference of the circle.

And pause here again to consider which of these statements explains why the line segment is not a radius of this circle.

And the answer is B, because the line segment should touch the circumference with one of its endpoints, and this one it does not.

And lastly, pause here to consider which of these statements explains why the line segment is not a diameter of the circle.

And the answer is A, because the line segment does not pass through the centre of the circle like this.

Here are some different parts of a circle, one that doesn't quite look like a diameter, and two which are two-dimensional parts of a circle, parts of a circle with an area.

Do they have names? As Aisha says, "They're definitely not the circumference, radius, or diameter," so what are they called and what are the specific details of their properties? Different parts of a circle can be split into two categories.

One, parts of a circle that have a length that are a distance of some sort, and two, parts of a circle that have an area.

The circumference, radius, and diameter all describe lengths as part of a circle.

There are also two more parts of a circle that describe a length.

One of these is a chord.

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

A chord does not need to pass through the centre of a circle like a diameter does.

The other is an arc.

An arc is a part of the circumference of a circle and is always a completely curved distance.

We can divide a circle up into different parts with each part having an area.

First of all, we can divide a circle into two parts using a chord.

One of these two parts is called a segment.

Each segment is a two-dimensional shape.

If the segment is under 50% of the area of a full circle, then it is called a minor segment.

You may have noticed that when you divide a circle using a chord, there are actually two segments.

If one is a minor segment, then the other is a major segment.

It has over 50% of the area of the full circle.

Oh, but what do you call a segment that is exactly 50% of the area of full circle? That's right, a semicircle.

If a chord divides a circle up into two semicircles, then the chord is also a diameter of that circle.

Another way of dividing up the space bounded by a circle is by dividing the circle up into sectors.

This is a sector.

A sector is formed by two radii and an arc joining them.

Note that the vertex of every sector must be at the centre of a circle.

Therefore, this is definitely not a sector as its vertex is not at the centre of a circle.

For this check, pause here to match up all four parts of these circles with their name.

And here are the answers.

One is an arc, two is a sector, three is a chord, and four is a segment.

For this check, we have the statement.

This is a sector.

Was that statement true or false? Pause here to choose a statement that justifies your answer of true or false.

And the answer is true because this part of a circle is formed from two radii and an arc, the definition of a sector.

And the next statement is, this is a segment.

Was that statement true or false? Pause here to choose a statement that justifies your answer of true or false.

The answer is false because a segment requires a chord to divide a circle into two parts.

Neither of the two line segments in this diagram are chords.

Parts of a circle can have either a length or an area.

If it has an area, then it is either a sector or a segment.

If it has a length, then it is either inside the circle and is therefore a radius, diameter, or chord, or on the perimeter of a circle and is therefore a circumference of a circle if it is the whole perimeter of the circle, or an arc if it is only part of the circumference of that circle.

Pause here to have a look at this summary of parts of the circle broken down into its different features.

Brilliant.

Onto the practise task.

For question one, pause here to match the part of the circle with its name.

And of questions two to five, pause here to read through each description and write down the correct names for each part of circle or the correct details of that part of a circle.

Great work so far on applying your knowledge and understanding to this range of information.

Pause here to compare your answers to question one with those on screen.

For question two, A describes a radius, B describes a sector, C describes either a diameter or a chord, as a diameter is a special type of chord, and D described a major segment.

For question three, here are some examples of different parts of a circle with an area.

And for question four, a segment is also a sector if the segment is formed from a chord that just so happens to also be the diameter of the circle.

This is because a diameter is made from two radii and two radii are part of what makes a sector.

Both the segment and the sector are therefore a semicircle.

And finally, question five.

The correct answer is 20 centimetres.

This is because the maximum length of a chord is always a diameter.

Now that we're more familiar with different parts of a circle, let's look at how we can find out either the length of the circumference of a circle or the area of a circle, starting with a square.

Let's have a look.

The perimeter of this square is four lots of six or four multiplied by six, which is 24 centimetres.

Andeep notices that this circle fits perfectly inside the square.

Andeep also spots that the circumference of the circle looks smaller than the perimeter of that square, but isn't sure how to show whether this is true or not.

We can calculate the length of the circle's circumference by considering pi, six multiplied by pi, in fact, where pi is approximately equal to 3.

14 and so the length of the circumference is 18.

85 after some rounding, so Andeep was correct.

The length of the circumference of that circle is smaller than the perimeter of the square.

Great observation, Andeep.

The perimeter of a square is four lots of the length of one of its sides.

On the other hand, the circumference or perimeter of a circle is pi or approximately 3.

14 lots of the length of the diameter of the circle.

Therefore, no matter how big or small a square is, if one of the sides of the square is the same length as the diameter of a circle, the circle's circumference will always be smaller.

4 d is always longer than 3.

14 d.

The circumference of a circle can be considered in two different ways, either in terms of its diameter or in terms of its radius.

If you are given the diameter of a circle, you can say its circumference is pi times by the diameter.

However, two lots of a radius is equal to one diameter.

Therefore, the circumference can also be considered rather than using the diameter as pi times two lots of the radius, also shown as two pi r or two times approximately 3.

14 times by r.

In summary, for a circumference C of any circle, we can use either one of these two formulae.

The circumference is equal to pi d or the circumference is equal to two pi r.

The length of the circumference of a circle can be written in two ways, either in terms of pi or as a rounded decimal.

For example, for this circle where we are given the diameter, I can use the circumference equals pi times d formula where d equals 18.

Leaving this in terms of pi means writing it as, for example, 18 pi, which implies 18 times by approximately 3.

14.

Conducting this multiplication of pi and the diameter will usually give a decimal, which we can round in this case to 56.

55 to two decimal places.

This is also true when given the radius rather than the diameter.

We can use the circumference equals two pi r formula, which in this case is two times pi times by a radius of 11, which can be simplified to 22 pi by multiplying together the 2 and the 11 or 69.

1 centimetres when rounded this time to three significant figures.

For this check, pause here to match each circle to the correct calculation that finds the circumference of that circle.

For option one, we are given a radius of eight centimetres, whilst for option two, we have a diameter of eight centimetres.

These are the appropriate formulae to use with each circle and so these are the correct calculations.

Next up, pause here to look through all of these calculations and values and choose the correct options that represent the length of the circumference of this circle.

Answer D shows the circumference equals pi d use of the formula, whilst B uses the C equals two pi r version of the formula.

F gives the answer in terms of pi at 24 pi, and H gives the correct answer, this time rounded to one decimal place.

Next up, we are given a calculation to find the length of the circumference for the circle.

This calculation is the circumference equals two times pi times 53.

Using your understanding of the formulae for the circumference for a circle, pause here to select the correct statements about this circle.

The formula being used is the two pi r version of the formula.

Therefore, we have an r radius of 53.

And therefore, the diameter is twice the radius at 106.

And once more, pause here to choose the correct statements for a circle whose circumference is 80 pi.

80 pi or pi times 80 is the pi d version of the formula.

Therefore, the diameter is 80 centimetres, the radius is half that at 40 centimetres.

The area of this square is six times six or six squared, which is 36 centimetres squared.

Andeep's observations are on point again.

The circle does have a smaller area than the square.

We can see this because the circle sits completely inside the area of the square.

However, it is difficult to show this fully using the diameter as there is no commonly used formula for the area that uses the diameter, only the radius.

So instead, let's break down the square into four congruent smaller squares, each with length three centimetres, to match the radius of that circle, also with three centimetres.

The area of that large square is therefore four lots of the area of one small square at three times three, or four lots of three squared, which gives us the same answer of 36 centimetres squared.

For the area of this circle, we can say that the area is pi times by the radius squared, so pi times three squared.

And because pi is approximately 3.

14, we can see that the area of the circle is about 28.

27 centimetres squared after some rounding.

The square has an area of four lots of three squared, whilst the circle has an area of 3.

14 lots of three squared, which therefore shows that the circle has a smaller area.

Finding the area of any circle is easiest when using the radius of the circle.

If you're given a diameter, just half it to get the radius.

The area A of any circle can be found using the formula, the area equals pi r squared.

For this check, pause here to match each circle to the correct calculation that finds the area of that circle.

For option one, we are given a radius of 10 centimetres, whilst two has a diameter of 10 centimetres.

For option two, this means that the radius is half of 10 at five centimetres.

Therefore, these are the correct calculations, both which use the radius of the circle.

For this next check, the area of this circle can be shown by the calculation, the area is equal to pi times 19 squared.

Pause here to use your understanding for the formula for the area of a circle to identify which of these statements are correct for this circle.

The area formula is pi times r squared, r for the radius.

Therefore, the radius of this circle is 19 centimetres.

The diameter is double that at 38 centimetres And once more, pause here to choose the correct statements for a circle whose area is 144 pi centimetres squared.

pi times 144 is pi times r squared, so 144 is r squared or the radius squared.

Therefore, r itself is the square root of 144, which is 12.

The radius is 12 centimetres.

Therefore, its diameter is double that at 24 centimetres.

It is sometimes possible to identify whether a calculation shows a circumference or an area of a circle even if the circumference or the area is not explicitly told.

For example, we have a calculation of two times pi times 45.

This is in the exact same form as the formula, the circumference equals two times pi times r.

Notice how they are both two times pi times something, a value.

This value is the radius of 45.

It is very common to see two times pi times something be the calculation for a circumference and so that something usually represents the radius of a circle.

Using this, we know the radius is 45 units, then the diameter is double that at 90 units.

The area is, therefore, 45 squared pi units squared, which after evaluation is 2,025 pi units squared.

Furthermore, what about the expression pi times 27 squared? This is in the exact same form as the formula, the area equals pi times r squared.

Again, notice how they are both pi times something squared.

This something is a radius of 27.

It is even more common to see pi times something squared be the calculation for an area and therefore that something usually represents a radius.

Again, if the radius is 27 units, then the diameter is double that of 54 units.

Its circumference is then two times pi times 27 units or 54 pi units if using the pi d version of the circumference formula.

For this final check, pause here to complete each sentence that describes Aisha's circle.

Because the calculation is 23 squared, it is most likely to calculate an area of a circle with radius 23 units.

Therefore, it has a diameter of double that at 46 units and a circumference of 46 pi units.

Brilliant.

Onto the final set of practise questions.

For question one, pause here to complete the table of information for each circle and leave your answers in terms of pi.

For question two, pause here to complete the table of information for each circle, but this time round each answer to one decimal place.

And finally, for questions three, four, and five, use the information provided to find out as much information as you can about each circle.

Pause now to do this.

Brilliant work.

Here are your answers for question one.

Pause here to compare your answers to those on screen.

And for question two, pause again to compare your answers to those on screen.

For question three, pause here to check all of the information on screen for each of these three circles.

For question four, Izzy's calculation is more likely to find the area of a circle because she is squaring a radius of 98 centimetres.

If the radius is 98 centimetres, then the conference is 615.

8 centimetres and the area is approximately 30171.

9 centimetres squared.

And finally for question five, if 400 pi found the circumference of the circle, then 400 is the diameter, and so 200 is the radius.

If, however, 400 pi found the area of a circle, then 400 is the radius squared and the square root of 400 is 20, so 20 is the radius.

Great work, everyone, on all of the effort that you've put in to circles today in our lesson where we have looked at definitions of multiple parts of the circle that have a length, the circumference, radius, diameter, arc, and chord, as well as parts of the circle that have an area, the segment and sector.

We've also seen that the circumference of a circle can be calculated using either its radius or diameter using one of these following formulae.

Either the circumference equals pi times d, or the circumference equals two times pi times r.

Furthermore, the area of a circle can be calculated using the formula, the area equals pi times r squared.

That's all from me for today.

Thank you all so much for joining me, but until our next maths lesson together, take care and goodbye.