Lesson video

Hello everyone, thank you for joining me, Mr. Gratton, for today's maths lesson.

During this lesson, we will look at methods to find the perimeter of composite shapes that include arcs of a circle.

Well, what is an arc? An arc is a part of a curve, and a circular arc is a part of a circle's circumference.

But before we look at arcs in the context of composite shapes, let's first look at different distances that we can calculate on a sector of a circle.

The circumference is the length around the outside of a full circle.

The circumference can be found using the formula two times pi times R, R for radius, or pi times D, D for diameter.

Both of these formulae can also be used as part of the calculation for a sector.

However, two times pi times R is usually, but not always, preferred as it is more common for the radius rather than the diameter to be given.

The most common exception to this is a semicircular sector whose diameter is just as likely to be given as its radius.

An arc length is the length of the curve part only of a sector.

For a circular sector, the arc is a fraction of the circle's circumference.

In comparison, a perimeter of a sector includes both the arc length and the two radii that form the straight line segment bounds of the sector.

This is because a perimeter is the full distance around the outside of a shape, starting and finishing in the same location as each part of the shape is traversed.

As you can see, the arc is congruent between the two sectors.

Whilst on the perimeter, two radii are also considered alongside the arc length.

The circumference of this full circle is 12 pi.

This arc is 1/3 the length of the circumference, because the sector that it belongs to is 1/3 the total area of the circle.

The arc length is therefore 1/3 of 12 pi, which is four pi.

However, the perimeter of this whole sector is the arc length plus the two radii, each of six centimetres.

For its total perimeter, written in terms of pi, we can add together the two sixes, and express the answer as four pi plus 12.

The four pi belongs to the arc length of the sector, whereas the 12 belongs to the two radii of that same sector.

Okay, onto a quick check.

Which of these correctly shows the circumference of this whole circle? Pause now, and select one calculation and one answer given in terms of pi.

The answer is two times pi times R, where R is the radius of 10.

Two times pi times 10 simplifies to 20 pi.

Sticking with the same circle, but this time a sector of the circle, match the arc and the sector to its corresponding length and perimeter calculations.

Some calculations of A to F do not match with either.

Pause here to look through all six of these options.

That arc is 1/4 the circumference of the whole circle, and 1/4 of 20 pi is five pi.

The sector is the same as the arc length of five pi, plus 10 twice, once for each of the two radii of the sector.

Therefore, its most simplified form is five pi plus 20.

Is Aisha's statement correct? She says, "If I double the size of an angle in this sector, the perimeter of the sector will also double." Well, let's have a look if she's correct.

The original sector has an angle of 40 degrees, which is 1/9 of a full circle.

The length of this arc will be the circumference of a full circle, two times pi times R, where R is 18 centimetres, multiplied by 1/9 as the arc is 1/9 the length of the full circumference of the circle.

The arc length is therefore, four pi.

And the perimeter is four pi plus 18 plus 18, 18 once for each radii of the sector, giving a total perimeter of four pi plus 36.

Or when given in decimal form, 48.

6 centimetres.

Now, let's double the angle to 80 degrees.

The sector is now 2/9 of a full circle, rather than the 1/9 of the sector we looked at before.

The arc length is 2/9 of 36 pi, which is eight pi.

Notice how, yes, the arc length has doubled as the angle has doubled as well.

However, does the full perimeter? Well, let's keep going.

The perimeter is eight pi plus the same 18, plus 18 again, because there are still two radii, both of 18 centimetres, that bound the sector.

The simplified answer to this is eight pi plus 36, or 61.

1 centimetres.

Therefore, Aisha is incorrect.

Whilst the arc length does double, the perimeter of the sector is less than double.

This is because the radius stays the same length.

Since two radii are part of the perimeter, that part of the perimeter isn't doubled, only the arc part is.

And so the total perimeter of a sector is less than doubled if the angle itself is doubled.

Next up, is Sofia's statement correct? She says, "The perimeter of the sector with radius of 18 centimetres will be double the perimeter of the sector with radius of nine centimetres." This question is different from the one that Aisha asked, as here the angle between the two shapes is the same, it is the radius that doubles instead.

Both sectors are 1/6 of a full circle.

And so for the smaller sector, the arc is 1/6 of two times pi times nine, which is three pi.

Its perimeter is then three pi plus nine plus nine, giving a total of three pi plus 18.

And for the biggest sector, its arc length is 1/6 of two times pi times 18, 18 rather than nine.

This gives an arc length of six pi, double the arc length of that smaller sector.

The perimeter of this bigger sector is six pi plus 18 plus 18, because 18 is the radius of this bigger sector.

This simplifies to six pi plus 36.

In this instance, Sofia is correct.

Both the perimeter and the arc length have doubled when the radius of the circle that the sector belongs to has doubled itself.

This is because both sectors are similar to each other, and so the arc length and perimeter are multiplied by the same scale factor as the radius.

Okay, onto the next few checks.

Each shaded region represents a different sized sector taken from the same circle.

How many times bigger is the angle of sector B compared to the angle of sector A? Pause here to consider how many times bigger 216 degrees is compared to 72 degrees.

Sector B has an angle three times bigger than sector A.

Sticking with the same two sectors, the arc length of sector A is 16 pi.

Using what Aisha and Sofia have discovered earlier, what is the arc length of sector B? Pause now to think about the relationship between these lengths.

And the answer is C, the arc length of a sector will triple if the angle of the sector is also tripled.

However, the next check is referring to the perimeter, not the arc length.

If the perimeter of sector A is 16 pi plus 80, what is the perimeter of sector B? Pause now to consider what has changed, and what has stayed the same between these two sectors.

The correct answer is B.

Whilst the arc length has tripled from 16 pi to 48 pi, notice how the radii have not changed length, just the orientation in the case of one of them.

Because the radii have not changed length, it is still the same plus 80 for sector B as it was for sector A.

And next check, sector A is the same, but now we are comparing it to sector C, which has a larger radius, but the same angle of 72 degrees.

What is the perimeter of this larger sector? Pause now to think about the relationship between all of these lengths.

And the answer is D, both the arc and the two radii will have doubled in length.

And so, the total perimeter will have also doubled.

Onto the practise, complete each statement with both a calculation and an answer, shown in the boxes on the right.

Pause here to complete each of these sentences.

And for question number two, complete the table by filling in the fraction of a full circle that each sector represents, an appropriate calculation for the arc length.

And then a calculation and answer for the perimeter of the sector.

I will give a signal to pause twice, once for these three sectors, and then once again for another three.

Pause now for parts A, B, and C.

And again for parts D, E, and F.

Decide whether each of these statements are true or false.

For any false statements, write a sentence explaining why you think it is false.

Pause now to look through all four of these statements.

For question number four, a sector has a radius of 72 centimetres and an angle of 70 degrees, its arc length is 28 pi.

Pause here to give yourself some time to answer these four questions about this sector, and other sectors related to this one.

And finally, for question number five, here are two pathways, each one is made from repeated congruent arcs.

A are semi-circular arcs, whilst B are quarter circular arcs.

Pause now to calculate the length of each of these two pathways.

Onto the answers, the circumference of the full circle is eight pi.

The arc length is 1/4 of that, at two pi, and the perimeter is two pi plus two lots of the radius of four, giving a total of two pi plus eight.

For question number two, pause here to compare your answers to the ones that are on screen.

And again for parts D, E, and F.

And for question number three, both A and C are true.

B is false because the radii do not triple, and D is false as the radii have also been divided by five.

And so, the correct answer is 14 pi plus 16, where the 16 is 1/5 of 80.

For question number four, the answers are 28 pi plus 144, 84 pi, 84 pi plus 144, and 14 pi plus 72.

For question number five, A, there are four semi-circular arcs with a total arc length of 28 pi metres.

And for part B, there are three quarter circular arcs, each with a radius of 23 metres, giving a total arc length of 108 metres.

Now that we've looked at the arcs in isolation, let's have a look at them in the context of a composite shape.

For the purpose of this lesson, we will only look at the perimeter of closed shapes.

If you travel along the perimeter of a closed composite shape, you will get back to the starting point without travelling across any part of the perimeter twice.

For the bottom non-example, this is not a closed shape, as there is no way of travelling across its length and returning to the beginning without traversing across part of its length more than once.

To calculate the perimeter of a closed composite shape, you can break it up into the lengths of its component parts.

Each component will either be a straight line segment, or an arc.

For example, this composite shape can be broken down into a semicircle, a quarter circle, and two line segments.

This time, each line segment is of a different length.

The length of the longer line segment is the 10 centimetres from the original diagram.

The diameter of the semicircle is three centimetres, and therefore, its radius is 1.

5 centimetres.

The radius of the quarter circle is three centimetres, and therefore, the remaining length of the shorter line segment is 10 take away three, or seven centimetres, as the shorter line segment plus the radius of that quarter circle is the same length as the longer line segment with total 10 centimetres.

We can calculate its perimeter by adding together each of the four lengths in turn.

Starting with the arc length of the semicircle at 1/2 of two times pi times a radius of 1.

5.

This gives us a total semicircular arc length of 1.

5 pi.

Then the arc length of the quarter circular arc is 1/4 times two times pi times a radius of three.

This also gives us a 1.

5 pi arc length.

The total perimeter is the sum of the two arcs, the 1.

5 pi plus 1.

5 pi, plus the sum of the two line segments at 10 and seven.

Therefore, the total perimeter all around that closed composite shape is three pi plus 17, or 26.

4 centimetres if converted into decimal form.

For this demonstration, I will show you a set of steps for the shape on the left and then prompt you to try the same set of steps for the shape on the right.

First of all, let's consider what component parts this shape is made from, and sketch each of the four component parts.

There are two straight line segments.

We know they are both of length 14 centimetres, because the hash marks mean they are of equal length.

There are also two semi-circular arcs, both have a diameter of 14 centimetres, and therefore, the radius of each is seven centimetres.

After labelling all relevant lengths, calculate the arc length of each arc.

As both arcs are congruent in this question, they are both the same, each with 1/2 the length of a full circle with radius of seven centimetres.

So 1/2 of two times pi times seven, which is seven pi each, or 14 pi altogether if you considered the two semicircles brought together to make one full circle.

To find the total perimeter of the shape, find the sum of all the arcs, plus the sum of all of the line segments, giving a total of 14 pi for the arcs, plus 28 centimetres for the line segments.

Pause now to try this yourself with the composite shape on the right.

And the answer for your shape should have been 22 pi plus 44.

It was also possible to treat your two semicircles as one full circle with a radius of 11 centimetres, saving you from having to halve the length of each arc to only add two of them together again later on.

Is Laura's statement true or false? Laura says "Both shapes have an area of 4.

5 pi plus 18." "Because both shapes have the same area, they will also have the same perimeter." Is it true that if two shapes have the same area, they will also have the same perimeter? Well, let's have a look.

For the first shape, I can break it down into four parts.

One semi-circular arc with an arc length of three pi, and three line segments, two with length three centimetres, and one with length six centimetres.

Each of these three line segments have lengths that can be added together, and therefore, the total perimeter of all four parts is three pi plus 12 centimetres.

In comparison, the second shape is composed of a quarter circle with an arc length of 1.

5 pi.

It also has a second congruent quarter circle, therefore, it also has an arc length of 1.

5 pi.

Trickier is calculating the length of the two line segments in this question.

Whilst six centimetres is labelled, this is not the full vertical length of that line segment.

There is an extra unlabeled length on the top left of the shape that is actually the length of the radius of that quarter circle.

And therefore, that vertical length has an extra three centimetres added to it.

This means that the two line segments both have a total length of nine centimetres each.

The total perimeter is therefore, 1.

5 pi twice, once each for the two arc lengths, plus nine twice, once each for the two line segments.

A total of three pi plus 18 centimetres.

Therefore, let's compare.

The first composite shape has a perimeter of three pi plus 12, whilst the second composite shape has a perimeter of three pi plus 18.

Laura is incorrect.

Just because two shapes have the same area, that does not mean they will also have the same perimeter too.

Some may, whereas some others will not.

Onto the final set of checks.

Which of these line segments and circular arcs are component parts of this composite shape on the left? Pause here to look at both the shapes involved, and the appropriate measurements on each component part.

Consider converting a radius into a diameter, or vice versa, when trying to figure which component parts match with this composite shape.

A is the horizontal line segment at the bottom of the shape.

D is the vertical line segment on the left of the shape.

F is the semicircle at the top of the shape, but with a radius of 3.

5 given, rather than a diameter of seven centimetres.

And G is the quarter circle on the right of the shape, sharing the same two centimetre radius as the vertical line segment.

Next check, how many copies are there of each component part in this composite shape? Pause now to look at each of the shapes in detail.

Each component part appears once, except for the two centimetre line segment, which appears twice.

Alternatively, you could have treated that bottom line segment as one longer line segment of nine centimetres, rather than separating it into two parts, the seven centimetre part and the two centimetre part on the very right.

And the last check, by considering the length of each line segment and arc individually to begin with, calculate the total perimeter of this shape, leaving your answer in terms of pi.

Pause here to perform all the calculations for this semicircle and quarter circle.

The answer is 4.

5 pi plus 11.

Onto the final set of practise questions.

For this composite shape, pause here to label each line segment and arc with the appropriate lengths.

Calculate the arc length of the semicircle and, hence, find the total perimeter of this composite shape.

For this composite shape, pause the video here to label each of the four line segments with their appropriate length.

Calculate the arc length of the two arcs by first calculating what fraction of a full circle the two arcs represent and, hence, find the total perimeter of that composite shape.

This composite shape has been broken down into its component parts.

Write down the length of each line segment, and note that some line segments will be a different length to others.

By identifying the radius of each quarter circular arc, calculate their arc lengths.

After calculating all arc lengths, find the total perimeter of the composite shape leaving, your answer in terms of pi.

Pause here to give yourself time to consider which lengths match with which parts of the shape.

And for question number four, starting with the smallest perimeter, sort these shapes in terms of the size of their perimeters.

Pause here to break down each shape into their component parts, if they have any, and consider the lengths of each part in order to find their total perimeters.

And finally, for question number five, Oak Field Academy have two running tracks, each with grass inside the track.

Each running track is made from two semicircular arcs plus two straights, made from line segments.

Show the Jun's statement about the two tracks is incorrect.

Pause here to read through Jun's statement, and figure out why he is incorrect.

And now, onto the answers.

The two vertical lengths are 10 centimetres each, and the horizontal length is eight centimetres, meaning that the diameter of the semicircular arc is also eight centimetres, therefore having a radius of four centimetres.

The total perimeter is four pi plus 28 centimetres.

Each of the four line segments is 15 centimetres each, for a total of 60 centimetres.

Each of the arcs has an arc length of six pi, and so the total perimeter is 12 pi plus 60.

The arc at the bottom of the composite shape has a radius of six centimetres, whilst the arc on the right has a radius of eight centimetres, giving each arc a length of three pi and four pi respectively.

The total perimeter of this composite shape is therefore seven pi plus 28 centimetres.

B is the smallest, with a perimeter of 20.

6 centimetres.

A is larger, with a perimeter of 25.

1 centimetres.

C is slightly larger still, at a perimeter of 26.

6 centimetres.

Whilst D is the largest, at 30.

3 centimetres.

For question number five, the perimeter of running track B is longer at a total of 202.

8 metres long.

However, the area of running track A is greater, at 2,463 centimetres squared.

Therefore, Jun's statement is incorrect.

The perimeter of B is longer, whilst the area of A is greater.

And a very well done on tackling some of those super challenging shapes and problems. That was the final question in today's lesson where we have covered the two lengths that can be considered from a sector of a circle, an arc length and a perimeter, and how the properties of each differ.

We've also considered how the length of an arc is proportional to the full circle that that arc represents, depending on the angle that the arc has.

We've also looked at strategies of breaking down a composite shape into component parts in order to find its perimeter.

Thank you so much for joining me for today's lesson on perimeters.

I hope to see you soon for some more maths fun, but for now, have a great rest of your day.