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 calculate the areas of different circular sectors.

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

First up, let's see how we can calculate the area of a sector.

To find the area of a circle, we use the formula pi times r squared.

Which for this circle is pi times 18 squared because the radius of that circle is 18? In terms of pi, we simplify this to 324 pi or when rounded is 1,017.

9 centimetres squared.

But what about this semicircle? What will the area of this shape be? Well, because this semicircle is from that full circle on the left, its radius will still be 18 centimetres.

In fact, the semicircle is exactly half of the circle on the left.

What is the area of this semicircle? Well, because this sector is half of a circle, its area is also exactly half the area of the full circle.

So its area is going to be 1/2 times by pi r squared, which is for this semicircle, 1/2 of pi times 18 squared or 162 pi when simplified.

For this first check, rather than a semicircle, we now have a quarter circle that is from this full circle with an area of 40 pi on the left.

Pause here to find the area of this quarter circular sector.

The area of this quarter circular sector is one quarter of the area of the full circle.

Therefore, 40 pi divided by 4 is 10 pi.

Next up, here's a check in two parts.

Pause here to find the area of this full circle.

The radius of this full circle is 9 and so 9 squared is 81, so the area of this circle is 81 pi.

Next up, this sector is one of three congruent sectors from the circle on the left.

Pause again here to find the area of one of these sectors.

Since this sector is 1/3 of a full circle, the area is going to be 1/3 of 81 pi, which is 27 pi.

It is possible to find the area of a sector with any angle.

We can do this using proportional reasoning so long as the radius of that sector is also known.

Here is the circle that this sector comes from.

We know this because both share a radius of 15 centimetres.

A full circle has an angle from its centre of 360 degrees.

The area of this full circle is, therefore, pi times r squared where R is 15, so pi times 15 squared, which when simplified is 225 pi.

For the sector of this circle, we have a 144 degree angle.

Therefore, the multiply from the angle of 360 degrees to the angle of 144 degrees is 144/360.

This fraction can be simplified, but if later on, you're gonna type in this calculation onto a calculator, there is no need to simplify this fraction.

For non-calculated questions however, simplifying this multiplier may be helpful.

Therefore, the area of this circular sector is 144/360 times by the area of the full circle, giving 90 pi as its area.

Leaving the area at the end of the calculation in terms of pi can be helpful to keep precision.

However, it is also possible to give the area in decimal form at approximately 282.

7 centimetres squared for its area.

For this next check, find the values of a to e in this ratio table in order to find the area of this sector.

I recommend filling in this table in alphabetical order.

Pause now to do this.

The angle for the full circle is always 360 degrees.

The area of the full circle is 16 squared pi or 256 pi.

Because the angle of the sector is 234 degrees, the multiplier from the full circle to the sector is 234/360.

And so to find the area of the sector, my calculation is 234/360 times by 256 pi, giving an area of that sector of approximately 481.

9 centimetres squared.

We can use a ratio table to find a general formula for the area of any sector.

This is really helpful as it allows us to find the area of a sector without drawing or considering the full circle that the sector came from.

Regardless of the size of the circle, it's always got a 360 degree angle from its centre and therefore, the area of any circle is going to be pi times r squared, for r being its radius.

For any sector with an angle of theta, theta is just any angle, the multiplier from 360 to theta is theta over 360.

And so applying that multiplier to the area formula over full circle gives theta over 360 multiplied by pi r squared.

Therefore, the general area formula for any sector with a radius of r and an angle of theta degrees is going to be this formula.

Notice how it is the area formula for a full circle multiplied by a fraction.

Great stuff.

Let's have a look at applying this formula to the sector on screen.

This sector has an angle of 306 degrees.

This means it is the fraction 306 out of 360 of a full circle where this full circle has an area of pi times r squared at a radius of 85.

Therefore, the area of this sector is 306/360 times by pi times by 85 squared.

Following the order of operations, we evaluate the square of 85 first, giving 7,225.

Typing this calculation into a calculator gives 19,293.

3 centimetre squared after some rounding as the area of this sector.

For this check, we have two methods to find the area of this sector with an angle of 120 degrees and a radius of 21 centimetres.

Pause here to consider whose method is correct, Andeep's or Sofia's.

And for the person whose method is incorrect, can you spot why? Sofia's method is correct.

Andeep's method is not quite correct because he hasn't followed the correct order of operations.

He has divided the radius of 21 by 3 before squaring the result.

However, as we can see with Sofia's correct method, the correct method involves squaring the radius of 21 first to give 441 and then dividing 441 by 3.

Brilliant.

Onto the practise questions.

For both questions one and two, find the areas of each sector.

For question two, you will first have to find the area of the full circle on the left of that question.

Pause now for questions one and two.

For question three, pause now to complete each table and write down a suitable multiplier to find the area of each sector.

And finally, for question four.

Starting with the smallest, pause here to put the sectors in order of the size of their area.

Brilliant effort on task A.

Onto the answers.

For question one, the areas were 30 pi and 40 pi or 94.

2 centimetre squared and 125.

7 centimetres squared respectively.

For question two, the area of the semicircle was 1,152 pi and the area of that other smaller sector was 384 pi.

Pause here to compare the information on screen to your calculations and answers for question three.

And finally for question four, the area of B was the smallest, then A, D, then C had the largest area.

Now that we're more familiar with how to find the area of a sector, let's see if we can use this knowledge to deepen our understanding and apply it to different contexts.

Right, first of all, let's have a quick look at this sector.

If we were to triple the angle of this sector from 43 degrees to 129 degrees, would the area of the sector also triple in size? Pause here to think about or discuss if Andeep's statement is correct.

Let's find out using ratio tables to show our thinking.

For this original sector, we have from its full circle an area of 20 squared pi or 400 pi.

And for this 43 degrees sector, we have as its area 43/360 times by 400 pi.

The area of this sector is approximately 150 centimetres squared.

However, for the sector after, we've tripled the angle.

Now at an angle of 129 degrees, we have an area of 129 over 360 times by 400 pi, which is approximately 450 centimetres squared.

We can see the angle of the sector has tripled in size and also the area of the sector has tripled in size as well.

Andeep's statement was correct.

We can generalise this relationship.

If you multiply the angle of a sector by any value k, then its area will also be exactly k times larger.

However, this is only true if the radius stays the same length.

Pause here to think about or discuss any circumstances where this may not be possible.

Right, for this check, both sectors A and B have the same radius, but we don't know what that radius is.

The area of sector A is 1,100 centimetres squared Pause here to calculate the area of sector B.

The angle of sector B is four times larger, so the area must also be four times larger at 4,400 centimetres squared.

Next up, Jun says, "If I multiplied the angle of that sector by 7, its area will be 7,700 centimetres squared." Pause here to think about why Jun is definitely incorrect with that statement.

A full circle can only have 360 degrees.

Multiplying the angle 52 degrees by 7 gives a total of 364 degrees.

Making the sector that Jun suggests actually impossible.

The sector just does not exist.

Andeep attempts to investigate a different theory.

This time suggesting that if the radius is tripled, the area will also triple in size.

Pause here to think about or discuss if Andeep is correct this time.

As with before, let's look at a ratio table to show our thinking.

A full circle with a radius of 14 centimetres has an area of 14 squared pi or 196 pi.

Therefore, the sector is a fraction of 196 pi at approximately 101 centimetres squared.

But now, here's our similar sector with triple the radius.

A full circle with a radius of 42 is pi times 42 squared or 1,764 pi.

Therefore, this sector is a fraction of 1,764 pi at approximately 908 centimetres squared.

The radius of the sector has tripled in size.

However, the area of the sector has definitely not tripled in size.

It's far, far bigger than triple in size.

The relationship between the radius and area is not a linear relationship.

Therefore, direct proportional reasoning does not work here.

For this check, is the following sentence true or false? The area of sector A is 100 centimetres squared, so the area of sector B is 400 centimetres squared.

Pause now to answer this question and justify your answer.

The answer is false.

Just because the radius is four times bigger, it does not mean the area is four times bigger.

In fact, the area is 16 times bigger at 1,600 centimetres squared.

Different pizza slices of different sizes may naturally cost a different amount of money, but Andeep asks whether it's possible to find the slice that is the best value for money.

We can consider which pizza slice is the best value for money by comparing its area to its price.

For this, single slice, small pizza, we can consider how much we pay for every one square inch of pizza that we actually get.

To do this, we must first calculate the area of the slice of pizza, which is a circular sector.

Because we are dealing with a sector, we can use the formula for the area of a sector where for this particular sector we have an angle of 50 degrees and a radius of seven inches, which gives 245/36 pi or approximately 21.

38 square inches.

So to find the cost per square inch, we take its total cost of one pound 50 and divide it by its total area.

This means for every square inch of pizza that we buy, we pay approximately seven pence.

Moving on, to the double slice pizza, we know its angle has been doubled and therefore, its area has also doubled at 245/18 pi or approximately 42.

76 square inches.

To find its total cost per square inch, we divide the total price of 2.

80 by its total area, giving a cost of 6.

5 pence per square inch.

Therefore, we are paying less per square inch of area of pizza in the double slice compared to the single slice.

We could also see this is true because the double slice is less than twice the price, but we get twice the pizza.

And finally, onto the single slice large pizza.

We still have a sector whose angle is 50 degrees, but this time with a radius of 14 inches, the area of this large single slice is, therefore, 50/360 times by pi times by 14 squared, which is different to the seven squared that we saw for the other two slices.

This gives an area of 245 over 9 pi square inches, which is approximately 85.

5 square inches.

To find its cost per square inch, we divide its total cost by its area to get a cost of approximately 7.

6 pence per square inch.

So going back to Andeep's original question, which pizza slice is the best value for money? Well, the double slice small pizza is the best value for money whilst the large pizza is the worst value for money.

For this final check, pause here to consider which pizza is the better value for money, the single slice or the triple slice? The triple slice has three times the area but is less than three times the cost.

Great work so far.

Time for the last practise task.

For questions one and two, complete each ratio table to show how the area of a sector changes as its angle or radius changes.

Pause now to do this.

And for question three, starting with the smallest, put these sectors in order of the size of their areas.

Pause now to do this question and see if you can make some calculations more efficient by using the linear relationship between angle and area.

And finally, question four, pause here to consider which pizza slice is the best and worst value for money, as well as other questions to do with these three pizza slices.

Great stuff.

Onto the answers.

For question one, the area of the smallest sector is 10/3 pi, which is doubled to 20/3 pi in the larger sector.

For question two, here are the two areas of those two sectors.

By dividing the area of the larger sector by the area of the smaller sector, we get a result of 36.

Showing that the larger sector is 36 times larger than the smaller one.

For question three, the correct order is D, A, E, F, C, then B.

Pause it to compare the answers that you calculated to the ones on screen.

And for question 4A, the area of pizza donated for one medium slice pizza is 22 square inches.

For part B, the area of one large slice is 50.

3 square inches, whilst the area of one small slice is 8.

7 square inches.

Meaning that the area of five small slices is 43.

6 square inches.

This replacement will have swindled the customer out of the equivalent of nearly one whole small slice of pizza.

And lastly, the medium slice is the worst value for money as you are getting the least amount of pizza for every one pound that you spend on it.

Great application of your knowledge of sectors in a lesson where we have used ratio tables to find the area of a sector by comparing the angle of that sector to 360 degrees.

We've also used ratio tables to show that the area formula is the area equals theta over 360 times by pi r squared.

We've also seen that multiplying the angle of a sector by k will also multiply its area by k as well.

However, multiplying the radius of a sector by k will not multiply its area by k because the radius and area do not have a linear relationship.

We've also used our understanding of proportional reasoning to compare how good or bad the value for money is for different slices of pizza.

I hope you're not too hungry after all of that pizza talk.

That's all from me though.

Thank you all so much for all of your attention.

Have a great rest of your day.

Take care and goodbye.