Lesson video

Hello, one and all.

I am Mr. Gratton, and thank you so much for joining me for today's maths lesson where we will look at composite shapes that have circle sectors as some of their component parts and look at strategies to find their area.

A sector is the region formed between the radii and the connecting arc.

Before we look at sectors as component parts of a composite shape, let's have a look at how we can calculate the area of a sector on its own.

The area of this circle can be calculated by pi times r squared where, in this case, r is 18.

We square the 18, giving 324.

So, in total, the area is 324 pi, when written in terms of pi, or 1,017.

9 centimetres squared if you convert it into decimal form and then round it to one decimal place.

But what if we had a semicircle instead? This sector is a semicircle that is exactly half of the circle on the left.

It was made by cutting along the diameter of that circle.

It has exactly the same radius as the circle.

If a sector is half of a circle, then its area is also half of the full circle that the sector came from.

Therefore, half of pi times r squared, in this case, is half of pi times 18 squared, which is 162 pi or 508.

9 centimetres squared when rounded.

Notice how 162 pi is exactly half of 324 pi, and 508.

9 is nearly half of 1017.

9.

That small difference is just down to the rounding between the two values.

You must always make sure that the correct order of operations is followed.

You must square the 18 first to get 324 before halving the 324 to get 162 for the total area of the semicircle to be 162 pi.

Each of these three sectors are congruent to each other and fit perfectly inside this circle with a radius of 6 centimetres.

This means that each of the three sectors makes up exactly one third of the full circle, which is 120 out of the full 360 degrees of each circle.

If we were looking at the area of this full circle, using pi times r squared, the area would be 36 pi.

However, if we are only looking at one third of the full circle, then its area would be 1/3 times pi times r squared, which, in this case, would be 1/3 of 36 pi or 12 pi centimetres squared.

What if the sector that we want to find out is actually two thirds of a full circle rather than one third from before? Well, we know that the area of a full circle is 36 pi, and 2/3 of 36 is 24, and therefore the area of this two thirds of a full circle is 24 pi or 75.

4 centimetres squared.

Okay, let's check your understanding.

Whose method to find the area of this quarter circle is correct, Andeep's or Sofia's? Pause to consider both methods, and find out which one is correct.

Sofia's is correct.

This is because you must square the radius first, in this case 12 squared is 144, before considering that a quarter of a full circle is going to be a quarter of that value, so in this case 1/4 of 144 pi.

In Andeep's, he divided by 4 first before squaring it which is the incorrect order of operations.

Onto the next check.

Which of these fractions correctly describes what fraction of a full circle this sector represents? Pause now to look through all five options.

Whilst both 270/360 and 3/4 are correct, 3/4 is the most simplified form and would be preferred over the 270/360 when doing the calculations to find the area of this sector.

Speaking of which, we are sticking with the same sector.

Which of these calculations correctly shows the area of this sector? Pause now to look through all three of these options.

a is the correct answer to the first step.

Pi times 20 squared would be the area of the full circle, and so 3/4 times by pi times 20 squared would be the area of this three quarters sector of a full circle.

20 squared is 400, and 3/4 of 400 is 300 which gives the second correct answer of c.

And for the final check, which of these calculations correctly shows the area of this semi-circular sector? Pay attention to the details in this question.

What's the same and what's different about the information provided here? Pause now to consider these details carefully.

In this question we are given the diameter of 10 centimetres rather than a radius which we have seen in previous questions.

If you are ever given the diameter, you must halve it to find the radius.

So, in this case, the radius of this semicircle is 5 centimetres.

Therefore the area of a full circle that this semicircular sector has come from is pi times 5 squared.

Because this semicircle is half of a full circle, the area can be expressed as 1/2 times pi times 5 squared.

Okay, onto some independent practise questions.

Here are three different sectors.

Match each sector with a calculation: either d, e, f or g; and an answer given in terms of pi: either h, i, j, or k.

Pause now to make these matches.

For question number 2, complete the table for each sector.

Step 1 involves writing a fraction times pi times radius squared, and then step 2 involves performing at least one appropriate calculation.

After doing the appropriate calculations, give your answer in the answer column both in terms of pi and in decimal form rounded to one decimal place.

I will pause twice, once for each slide that provides a table.

Pause now for these three sectors.

And pause here for these next three sectors.

And here are the answers for question 1 where a is a quarter circle, b is a semicircle, and c is a third of a circle.

And for question number 2, pause here to compare your answers with those in this first part of the table.

And pause again to compare your answers with the second part of the table.

And now that we've looked at sectors of a circle in isolation, how do we find the areas of composite shapes that include these same types of sectors? Well, there is a massive range of shapes that include these sectors, but, for today, we will only consider shapes that contain rectangular parts alongside its circular sectors.

We will look at three approaches to identifying the component parts of a composite shape with the first method looking at breaking a shape up into its separate component parts such as with this shape where we can split the shape up into a semicircle and a rectangle.

Notice how currently the semicircle isn't labelled with any dimensions.

Well the 10 centimetre horizontal width of the rectangle is exactly the same as the diameter of this semicircle.

And now that we've labelled all of the relevant dimensions on both of the composite shapes, we can consider the area of each of these parts separately.

For the semicircle, its diameter is 10 centimetres, and so its radius is half of that, at 5 centimetres.

Substituting 5 into the area of a circle formula and then halving the total, because this is a semicircle not a full circle, will give you 12.

5 pi centimetres squared.

The rectangle part is very straightforward.

The base of 10 times by the height of 8 is an area of 80 centimetres squared.

We can leave our answer in terms of pi.

This can be expressed by writing down the addition of the two terms, the pi term and the non-pi term, and leaving each of these terms separate; 12.

5 pi, and separately, plus 80.

On the other hand, we can use a calculator to convert this number into decimal form at 119.

3 centimetres squared.

For this demonstration, I will demonstrate how to find the area of a composite shape on the left, and, after each set of steps, apply those same instructions to the composite shape that you can see on the right.

Step 1 is to split the composite shape up into its component parts: a semicircle and a rectangle.

The semicircle has a diameter of 12 centimetres, and therefore its radius is half of that, at 6 centimetres.

And so, in the area calculation, we use r = 6 in pi times r squared to get 36 pi and then half it, because we're working with a semicircle, to get a total area of 18 pi for the semicircle.

The rectangular part is 3 times 12 which equals 36.

The answer can be given as either the addition of two terms: 18 pi and, separately, plus 36; or as a decimal form of 92.

5 centimetres squared.

Using the method that you see on screen, try finding the area of the composite shape on the right-hand side.

Pause to try this now.

And so the method and the answer to find the area of this composite shape is as follows.

A second way to find the area of a composite shape is by removing combinations of quadrilaterals and sectors of circles from a larger quadrilateral sector and then setting up a subtraction between the areas of its component parts to find the area of the resultant composite shape.

With this composite shape, imagine the entire shape as a rectangle with a quarter of a circle subtracted from it.

The radius of this quarter circle is the missing part of the height of the rectangle, giving it a total height of 30 centimetres.

As with before, we calculate the area of each part separately: the rectangle at 240 centimetres squared and the sector at 16 pi centimetres squared.

The area of the shaded region on the far left of the screen is the area of the rectangle, at 240, with the area of the quarter circle of 16 pi subtracted from it.

Onto the next demonstration.

As with before, after I show you each set of steps on the left, apply those same set of instructions to the composite shape on the right.

The first step should always be to separate the composite shape into its component parts.

In this case we have a rectangle with a semicircle and a quarter circle subtracted or removed from the rectangle.

Pause here to give that a go for your composite shape.

And for my composite shape on the left, the height of the rectangle is 12 centimetres.

And so this is the same as the semicircle's diameter.

The 12-centimeter diameter is the exact same as a 6-centimeter radius.

The quarter circle's radius is also 12 centimetres, and because it is a radius already given, we do not need to do anything with it.

Pause here to label all possible sides of your composite shape on the right.

We can now find the area of each composite part.

The rectangle has an area of 300 centimetres squared.

The semicircle has half the area of a full circle of radius 6 centimetres, which gives you 18 pi centimetres squared, and the quarter circle is a quarter of the area of a full circle of radius 12 centimetres giving an area of 36 pi centimetres squared.

Pause here to find the area of all three of your component parts.

The area of the shaded region is the full circle with both the semicircle and the quarter circle subtracted from it.

This then simplifies to 300 take away 54 pi.

Pause here to form the exact same subtraction to calculate the area of your shaded region leaving your final answer in terms of pi.

And here are the answers for the area of each composite part as well as the total area of the shaded region, which is 44 take away 6 pi centimetres squared.

And the final method for calculating the area of a composite shape is by rearranging some of the component parts so that the combination makes a more straightforward shape to calculate such as two semicircles being rearranged into a full circle or, in this diagram, four quarter circles, which can be rearranged into one full circle plus the extra square that existed at the centre of the original composite shape.

Okay, onto a check.

For this composite shape, which of these are component parts and rearrangements? Pause here to have a look at not just the shape of these parts but also the length given and whether these lengths represent a radius or a diameter of a sector.

And here are the answers: a, c, and f, where c has been formed by rearranging the two quarter circles into one semicircle with a radius of 5 centimetres.

Okay, onto the practise.

For question number 1, for both shape 1 and shape 2, match each one to the correct component parts as seen in the options a to f, there'll be multiple correct options for each shape, as well as the correct calculation in the options from g to j.

There'll be one correct option for each shape.

Pause now to give this a go, and match all correct options to both shapes 1 and 2.

And onto question number 2.

Complete this table by drawing each shape being broken down into its component parts and then labelling the lengths of each of the component parts that you drew.

After that, show a calculation to find the area of each component part, and by collecting the individual areas you've calculated, come to a final answer for the area of each composite shape giving your answer both in terms of pi and as a decimal that has been rounded to one decimal place.

I will pause twice, once for shapes a and b and again for shapes c and d.

Pause now for shapes a and b.

And again for parts c and d.

For question number 3, what do you notice about each shaded region? Show values and calculations to support any observations that you make.

Pause now to look at all three parts to this diagram.

And for question number 4, calculate the shaded region for each of these three shapes that are only made from rectangles and sectors of a circle.

Pause now to first label all possible lengths.

Rearrange some of the component parts.

And leave your answers in terms of pi.

And for question number 5, calculate the area of each shaded region.

The centre of each circle and each sector is labelled by a cross.

Pause now to do this final question.

And here are the answers to question 1.

Shape 1 matched with c and d, and the calculation I whereas shape 2 matched with e and a and calculation g.

And for question number 2, pause here to compare the shapes and values in your table with the ones that you see on screen.

And pause here to have a look at the answers for part c and d.

And an observation to make about question number 3 is that the area of each shaded region is actually equal.

The empty space in each shape can be rearranged to create one full circle, and so the area of each shaded region is what's left over when you have a square with a circle subtracted from it.

The area of each shaded region is therefore 64, the area of a square, take away 16 pi, which is the area of these three congruent circles just split up into different parts.

And for the first diagram of question number 4, its component parts are two quarter circles and one semicircle which can be rearranged into one full circle with an area of 400 pi centimetres squared.

For the second diagram, it has component parts of a rectangle, a quarter circle, and a quarter circle that has been subtracted away from the rectangle.

Therefore, you can move that extra quarter circle into the empty space left by the subtracted quarter circle, leaving a rectangle with an area of 5,400 centimetres squared which is just 45 times 120.

And for this third diagram in question number 4, there are nine congruent component semicircles.

Each of the semicircles has a diameter of 40 centimetres and therefore a radius of 20 centimetres.

Using the radius of 20 centimetres for each of the nine semicircles, each semicircle will have an area of 200 pi centimetres squared.

If one semicircle has an area of 200 pi, then the nine semicircles brought together will have a total area of 1,800 pi centimetres squared.

And for the first diagram of question number 5, it has an area of 128 take away 32 pi.

The second shape has an area of 2 pi, whilst the third area is 75 pi, and the fourth area is 108 pi.

Each of these should be given with the units centimetres squared.

And thank you for all of your amazing and hard work in a challenging lesson where we have found the areas of sectors of the circle using multiplicative reasoning and where we have found the area of composite shapes by considering the breaking, completing, or rearranging of component parts of each composite shape.

That is all for today's lesson.

I appreciate you for joining me here.

I hope you have an amazing rest of your day, and I hope to see you soon for some more maths.

Have a good day.