Lesson video

Hello everyone.

Welcome and thank you for joining me in today's lesson on 2D shapes.

Hi, I'm Mr. Gratton, and today we will compare arcs and perimeters of sectors and find the lengths of both.

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

First up, let's have a look at the difference between an arc and a perimeter.

If the length around the outside of a circle is called the circumference with a length of 2 pi r, then an arc is a fraction of that same circle's circumference.

It is the curved part of a sector.

On the other hand, a perimeter of a sector includes both the arc and two radii that form the bounds of the sector, like so.

Here's our arc and here is our sector with a perimeter.

That perimeter is made of one, two radii of equal length and an arc with a length.

For this full circle, it's circumference is 2 times pi times six, or 12 pi.

This arc is from one of those three congruent sectors that is one third of the length of the full circumference of that circle, and therefore the arc length is one third of the full circumference of 12 pi, which is 4 pi.

On the other hand, we have a perimeter of the sector that has two radii, each of 6 centimetres that enclosed the sector making the perimeter 4 pi plus a radius of 6, plus a radius of 6, or 4 pi plus 12, where the 4 pi is the length of the arc and the 12 belongs to the two radii, each at 6 centimetres.

For this first check pause here to consider which of these correctly shows the circumference of this whole circle.

A formula for the circumference is 2 times pi times r, which in this case is 2 times pi times a radius of 10, which when simplified gives you 20 pi.

For this next check we have a sector from this circle.

The circle has a total circumference of 200 pi.

Pause here to find the arc length of that sector.

That sector is one quarter of the full circle and so the arc length is one quarter of 200 pi, which is 50 pi.

We know that this sector has an arc length of 14 pi, pause here to consider what the perimeter of this sector is.

The perimeter is the arc length of 14 pi plus 15 twice for the two radii.

Therefore we have either 14 pi plus 15 plus 15 or 14 pi plus 30.

14 pi and 15 are not like terms so you cannot add them together, however, you can collect together the two fifteens to make 30.

And finally for this check, both the arc and the sector are from this circle with a circumference of 20 pi.

Match the arc and the sector to their corresponding length and perimeter calculations.

Some calculations do not match with either.

Pause now to do this.

The arc is one quarter of 20 pi, which is 5 pi in length, and so the sector is 5 pi plus the two radii, each at 10 centimetres.

We can find the arc length or perimeter for any sector from a circle by first of all considering the angle made by the two radii of that sector.

So we know that a full circle has 360 degrees from its centre.

Therefore, to radii from a sector make an angle of, well we don't know, so let's call it theta degrees.

This sector is therefore theta over 360 of a full circle and the arc of a sector has a length that is theta over 360 of the circumference of that full circle.

So for a sector with any angle, let's call it theta, and any radius, r, we can formalise the arc length of that sector through this formula.

The fraction, theta, over 360 times by the circumference of the full circle where the circumference of the full circle is 2 pi r.

This formula finds the arc length of any sector.

Note that we use the radius rather than the diameter as it is more common to see a radius labelled on a sector than a diameter.

The one most common time that you'll see a diameter over a radius is if we are given a semicircle.

Next up, what about the formula for the perimeter of a sector? We have the arc length, which we found out on the left plus the two radii each at length r.

Therefore the perimeter of this sector is theta over 360 for the fraction of a full circle times by 2 pi r for the circumference of the full circle, plus two lots of the radius r.

Right, let's have a look at these formulae in action.

This sector is from a circle with a radius of 40 centimetres.

Therefore the circumference of this circle is 2 times pi times 40, which equals 80 pi.

We can present our calculations and thinking for this sector in this table of information, the fraction of a full circle is the angle 162 over 360, because this sector has an angle of 162 degrees between its two radii.

The length of the arc of this sector is therefore the circumference of the full circle at 80 pi multiplied by this fraction, which when simplified gives us 36 pi.

The perimeter of the sector is the arc length of 36 pi plus two lots of the radius with each radius at 40 centimetres.

Therefore the perimeter of the sector is 36 pi plus 80.

Furthermore, it is also possible to give arc lengths and perimeters as rounded decimals.

However, it is helpful to first leave the circumference, the full circle, in terms of pi, this avoids multiple rounding errors when using the circumference to find the arc length or perimeter.

For this sector, we consider the full circle as 2 times pi times the radius of 11, giving 22 pi.

The fraction of a full circle that this sector represents is 263 over 360 because the angle between the two radii is 263 degrees.

The length of the arc of this sector is therefore the circumference of the full circle at 22 pi multiplied by this fraction.

When tightening this into a calculator, we get, after rounding, 50.

5 centimetres.

The perimeter of the sector is the arc length plus two lots of the radius, in this case, with each radius at 11 centimetres.

Therefore the perimeter of the sector is 50.

5 for the arc length plus two lots of 11, or a total of 72.

5 centimetres.

Let's break down these calculations in a few checks.

For this check pause here to consider what fraction of a full circle this sector is.

The fraction involves two angles.

The numerator is the angle of this sector, 138 degrees, whilst the denominator is the angle of a full circle from its centre, 360 degrees, and therefore for this same sector, which of these calculations shows the arc length of this sector? Pause now to choose the correct one.

The circumference of the full circle is 2 times pi times 65, or 130 pi.

130 pi is then multiplied by the fraction of a full circle, which we saw to be 138 over 360.

We could then type this calculation in to a calculator to find the arc length of this sector.

And finally pause here to match the sector with the correct calculation to find its arc length.

Remember, with each calculation you need to first correctly consider the circumference of the full circle as 2 times pi, times r, or twice the radius of that sector multiplied by pi.

Great stuff, onto the practise questions.

For question 1, the arc and sector are congruent to that sector inside that circle.

Pause here to complete each statement using a calculation and answer shown on the right of the screen.

For question 2, pause here to complete the table, leaving your final answers in terms of pi.

Question 3 is similar but give each arc length and perimeter as decimals to two decimal places.

Pause now for this question.

And finally, question 4 and 5, pause here to do some of these circle problem questions.

Great work in all of these circle questions so far, onto the answers.

For question 1, the circumference of the full circle is 2 times pi times 12, which simplifies to 24 pi.

The arc length is therefore one quarter of 24 pi at 6 pi.

The perimeter of this sector is 6 pi plus twice the radius each at 12 centimetres for a total of 6 pi plus 24.

And now pause here to compare your answers to question 2 to the ones on screen.

And again, pause here for question 3.

For question 4a, you are given an arc length of 64 pi and a radius of 160.

A perimeter of a sector is the arc length plus the radius plus the radius again, in this case, 64 pi plus 160 plus 160, or 64 pi plus 320.

For part b, sector Q is 3 over 16 of a full circle.

The arc length of Q is therefore three sixteenths of 2 pi times 128, which simplifies to 48 pi.

The perimeter is therefore this 48 pi plus two lots of 128 or 48 pi plus 256.

And lastly, question 5.

Sector A is longer, with a perimeter of 154.

2 compared to 142.

8 centimetres.

Next up, let's bring in some of our advanced knowledge of triangles, Pythagoras' theorem, and trigonometry, to help us with some sectors.

It's possible to find the arc length of a sector given only the length of a chord, as long as that chord has the endpoints that touch the endpoints of the two radii of that sector.

In the special case where the sector is a quarter circular sector, we can find the arc length using Pythagoras' theorem.

This is because in this specific case the chord divides the sect up into a segment and a right-angled triangle.

All of the radii in one circle are the same length.

This is therefore also true for any sector where both radii within one sector are both the same length.

Therefore, for this sector, both the radii can be labelled as r centimetres for some unknown distance, r centimetres.

So we can use Pythagoras' theorem on this right-angled triangle to give us 1 radius squared plus the other radius squared equals the hypotenuse squared, where the hypotenuse in this situation is the length of the chord, so in this case we have 32 squared.

This simplifies to 2 r squared equals 32 squared, where 32 squared equals 1024.

Dividing both sides by 2 gives us 1 r squared equals 512, square-rooting 512 gives us approximately 22.

6 centimetres, or given in surd form, 16 times by the square root of 2.

We can then use this radius to find the arc length or perimeter of that sector.

Sometimes it is helpful to use the srurd form of the radius rather than the rounded decimal form, but brace yourself, it might look a little bit complicated to begin with.

For the arc length we have 1 quarter because we are dealing with a quarter circle of 2 pi r, where r, the radius, is 16 root 2.

When we simplify all of this, we can express it in surd pi form as 8 pi times by root 2.

The perimeter is therefore 8 pi root 2 plus twice the radius, giving 8 pi root 2 plus 32 root 2.

Whilst this looks really complicated, this is just an exact form that is more precise than using the 22.

6 centimetres for the rounded version of the radius.

For this check, find the length of the radius, r, of this sector leaving your answer surd form.

Pause now to use Pythagoras' theorem to do this.

Setting up Pythagoras' theorem gives us r squared plus r squared equals 60 squared, which simplifies down to r squared equals 1,800, square rooting 1,800 gives us r equals 30 root 2.

So now that we know that the radius is 30 root 2, pause here to find out which of these calculations shows the arc length of that sector.

We are dealing with a quarter circle, and so a quarter of the circumference of a full circle is 2 pi r, where r in this case is 30 root 2.

So the arc length of this sector is a quarter times by 2 pi times by 30 root 2 which simplifies down to 15 pi root 2.

The problem is though not all sectors are quarter circles.

When the sector is not a quarter circle, we can find the arc length using trigonometry.

The triangular part of this sector is an isosceles triangle, due to both radii being the same length and one side having the length of the chord, we can split this isosceles triangle into two congruent right-angled triangles like so.

So by dividing this isosceles triangle into two right-angled triangles, the angle of 110 degrees and the length of 50 centimetres from that chord will both be halved.

Therefore, the angle of one of those two right-angled triangles is 55 degrees, and the length of the side opposite that 55 degree angle is 25 centimetres.

We can use the sine formula to find the radius of that sector.

The sine formula is the hypotonus times by the side of our angle equals the length of the side opposite that angle.

So for this right-angled triangle, the hypotonus is the radius of the sector r.

The angle is 55 degrees and the side opposite 55 degrees is 25 centimetres.

We want to solve for r to find the radius of this sector, and therefore we divide through by sine 55, giving r equals 25 over sine 55, which is approximately 30.

5.

The radius of this sector is therefore 30.

5 centimetres.

As with before, we now know the radius of this sector and therefore the arc length and the perimeter can both be found.

However, don't let the angle catch you out.

We must use the original angle, not the halved angle used in the trigonometry calculation.

The arc length is therefore 110 over 360, not 55 over 360 times by 2 pi r, where r is that radius, 30.

5, giving an arc length of 58.

6 centimetres.

Right, for this check we have an isosceles triangle that is split into two congruent right-angled triangles with this being one of those two right-angled triangles.

Pause here to use the information in that sector to find the values of a and b in that right-angled triangle.

A degrees is half of the angle 64, which is 32 degrees, and b centimetres is half of the length 24 centimetres, which is 12 centimetres.

Now we know that one of the angles in that right-angled triangle is 32 degrees, and the side opposite that angle is 12 centimetres long, pause here to consider which of these trigonometry equations correctly calculates the length of the radius of this sector.

The hypothesis of r multiplied by the sine of 32 degrees equals the side opposite 32 degrees, which is 12 centimetres long.

Next up, pause here to rearrange that equation to find the value of r.

We divide both sides by sine 32, so r equals 12 divided by sine 32, which is approximately 22.

64.

Therefore, the radius of this sector is 22.

64 centimetres.

Now pause here to identify which of these calculations finds the arc length of this sector.

Remember, don't Forget to use the original angle of 64 degrees, not the halved angle of 32 degrees that you used in the trigonometry calculation.

Brilliant, onto the final few practise questions.

For question 1, pause here to use Pythagoras' theorem to find the radius of this quarter circle and hence find the arc length of that quarter circle.

Next up, for question 2, you are given one of the two, congruent right-angled triangles made from drawing the perpendicular that splits this isosceles triangle into two.

Pause here to label as much information as you can on this right-angled triangle and then find the arc length of that sector with an angle of 116 degrees.

And finally, question 3.

We have three different sectors.

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

Amazing effort, everyone.

For the answer to question 1, the radius of the sector is 4 root 2, meaning its arc length is 1 quarter times by 2 times by pi times by the radius of 4 root 2, which simplifies to 2 pi root 2.

And for question 2, the angle atop that right-angled triangle is 58 degrees at half of 116 degrees.

The side opposite that angle has a length of 19 centimetres, half of 38 centimetres.

We can then create a trigonometry equation, r times by sine 58 equals 19, dividing both sides by sine 58 gives r equals 22.

4.

The radius is 22.

4 centimetres.

The arc of the sector is therefore 45.

4 centimetres long.

And finally, pause here to compare all of your calculations and perimeters that you found to the ones on screen.

The correct order was b, then a, then c.

Thank you all so much for all of your amazing work in a really challenging lesson where we found the arc lengths by considering a fraction of the whole circle that a sector came from.

We also considered that the perimeter of a sector is simply the arc length plus the two lengths of the two radii that bound that sector.

And remember, each radius of a given sector of a circle is equal in length.

The arc length of a sector can also be found by using either Pythagoras' theorem or trigonometry if given the angle of the sector and the length of a chord.

Once again, I appreciate all of your effort in today's lesson.

I've been Mr. Gratton, and so, take care and have an amazing rest of your day.