Lesson video

Hello, my name is Dr.

Rowlandson and I'm thrilled that you're joining me in today's lesson.

Let's get started.

So welcome to today's lesson from the unit of geometrical properties with polygons.

This lesson is called interior and exterior angles of regular polygons.

And by end of today's lesson we'll be able to find the interior and the exterior angles of regular polygons.

Here are some previous keywords that you may have come across before and we'll be reusing these words during today's lesson.

So you might wanna pause the video at this point while you re-familiarize yourselves with the meanings of these words before pressing play to continue.

This lesson contains three learn cycles.

In the first learn cycle, we're gonna be looking at exterior angles on regular polygons.

In the second learn cycle, we'll be looking at interior angles of regular polygons and then in the third learn cycle, we'll apply what we've learned to find missing angles on regular polygons.

But let's start by looking at exterior angles in regular polygons.

Let's start by reminding ourselves what a regular polygon is.

A regular polygon has sides that are all equal length and also has interior angles that are all equal too.

Now, if all the interior angles on irregular polygon are equal, what does that mean by the exterior angles? It means that the exterior angles of irregular polygon are all equal too.

Now, the exterior angles in any polygon always sum to 360 degrees.

Therefore, the exterior angles in irregular polygon can be calculated by dividing 360 degrees by the number of vertices that'll work out the size of each exterior angle.

For example, with this regular pentagon, if I wanted to work out the size of each of those exterior angles, well the sum of the exterior angles is 360 degrees.

There are 5 exterior angles on a pentagon.

So the size of each exterior angle would be 360 degrees divided by 5 to get 72 degrees.

So if dividing 360 degrees by the number of vertices or the number of sides gives us the size of each exterior angle on the regular polygon, then we could rearrange that calculation and think about how the number of sides of a regular polygon can be calculated when these sides of the exterior angles are known.

For example, a regular polygon has exterior angles of 45 degrees.

Can we work out how many sides does it have? Pause the video while you think about this and press play when you're ready to work through it together.

Well, Alex is gonna help us with this one.

Alex says, the exterior angles sum to 360 degrees.

We know that each exterior angle is 45 degrees, so we need to work out how many times does 45 go into 360? We can in calculate that by doing 360 divided by 45 to get 8.

So there's 8 sides.

It's an octagon and it's a regular octagon, so it looks something like this.

Let's check what we've learned there.

A regular decagon has 10 sides.

What is the size of each of its exterior angles? Pause the video while you work this out and press play when you're ready to continue.

Let's now work through this together.

The sum of the exterior angles in any polygon is 360 degrees.

Because this is a regular decagon it has 10 exterior angles.

So to get the size of each exterior angle, we're gonna do 360 degrees divided by 10 to get 36.

A regular polygon has exterior angles of size 30 degrees.

How many sides does it have? Pause the video while you have a go at this and press play when you're ready for the answer.

Let's work through it together now.

We know that the sum of exterior angles in any polygon is 360 degrees.

And we know that each exterior angle in this polygon is 30 degrees.

So to work out the number of sides or the number of (indistinct) that this polygon has, we can do 360 divided by 30 and that'll give us 12.

Therefore, this shape has 12 sides and is a dodecagon and would look something like this.

Here we have Sofia.

Sofia is using GeoGebra to programme a turtle to draw a polygon, and this is what she's got planned so far.

She's gonna programme the turtle to move forward three units, then turn left 50 degrees and then repeat those two instructions a certain number of times.

Sofia says, "If I keep repeating the same two instructions, all the sides will be equal and all the angles will be equal." And Sofia reckons it should draw a regular polygon.

What do you think? On the slide deck at the bottom of this slide, there is a link to a GeoGebra activity similar to the one that Sofia is using.

If you want to and are able to, you could pause this video, load it up and have a go at it yourself before pressing play to continue doing it together.

But let's see what Sofia does.

Sofia repeats these two commands seven times and this is what it draws.

Huh? It's not quite a polygon is it? Sofia says, "I've drawn seven sides so far.

The turtle is nearly back to the start, but there's a small gap." So if repeating these commands seven times isn't enough to close up the polygon, what would happen if we repeated those instructions eight times? Sofia repeats the two commands eight times and it draws this, huh? Again, that's not quite a polygon is it? Sofia says, "When I draw eight sides, the line segments overlap.

Why can I not get it to draw a regular polygon with a 50 degree turn?" Why do you think that is? Pause the video while you think about this and press play when you're ready to discuss it together.

We know that the exterior angles on any polygon sum to 360 degrees.

So if we think about how this turtle with these commands will be turning an exterior angle of 50 degrees each time, if we added up those exterior angles as we went along, we would have 50 or 100, 150, 200, 250, 300, 350 and 400.

We wouldn't get to 360 degrees if we keep using an exterior angle of 50 degrees.

So what can we learn from that? When a polygon is regular, 360 is a multiple of the exterior angle, therefore dividing 360 by the exterior angle would give an integer.

So if we look at what Sofia was trying to draw, we know that the sum of exterior angles in polygons is 360 degrees.

Each exterior angle that Sofia was trying to use was 50 degrees and if we divide 360 by 50, we don't get an integer, we get 7.

2.

A polygon cannot have 7.

2 sides, so a regular polygon cannot have exterior angles of 50 degrees.

So let's apply what we've learned there with this problem.

The two diagrams each show a single vertex of a polygon.

We can only see one vertex and the wobbly line just indicates that there is more to this polygon, but we just can't see it.

An exterior angle for each polygon is shown.

The one the left shows 40 degrees, the one on the right shows 35 degrees.

Determine whether it is possible for each polygon to be regular.

Perhaps pause the video at this point and consider how you might determine this and then press play to continue.

Let's start with the diagram on the left with the exterior angle of 40 degrees.

If we divide 360 by 40, we get 9.

Now that shows us that 360 is a multiple of 40.

Therefore the shape could be a regular polygon with 9 sides.

It could be a regular nonagon.

We don't know for certain that it's regular because we don't know what all the other exterior angles are.

They would all have to be 40 for it to be regular, but we know that it at least could be regular.

How about the diagram on the right? If we divide 360 by 35, we get 10.

285 and some more decimals.

That means it's not an integer, therefore 360 is not a multiple of 35 and the shape cannot be regular.

Let's check what we've learned so far with this.

True or false, regular polygons can have exterior angles of any size.

Is that true or is it false? And then choose one of the justifications below.

Pause the video while you make some choices and press play when you're ready for an answer.

The answer is false because 360 must be a multiple of the exterior angles in order for the polygon to potentially be regular.

Here's another question.

Can a regular polygon have exterior angles of size 80 degrees? Pause the video while you work this out and press play when you're ready for an answer.

Well, if we do 360 divided by 80, we get 4.

5.

4.

5 is not an integer, so the answer is no because 360 is not a multiple of 80.

Over to you now for task A.

This task contains 5 questions and they're all available here on the screen.

Pause the video while have a go at these and press play when you're ready for some answers.

Well done with that.

Let's now work through these 5 questions together.

In question one it says find the value of X.

We've got a regular hexagon, so let's do 360 divided by 6 to get 60.

In question two, a regular hexacontagon has 60 sides.

What is the size of each exterior angle? So let's take 360 divided by 60 and we get 6.

And then question three, a regular polygon has exterior angles of size 2 degrees, how many sides does it have? So if we take 360 and divide it by 2 degrees, we get 180, so it has 180 sides.

And question four, Aisha draws a pentadecagon, which is a polygon with 15 sides.

One of the exterior angles is 24 degrees.

Could it be a regular polygon? Explain your reasons.

Well, we know that the exterior angles in any polygon sum to 360 degrees, so any solution we give here should really start by stating that fact because it's what we're working with.

Exterior angles in a polygon sum to 360 degrees.

Once you've stated that we could write down 360 divided by 15 equals 24, that shows if we take the 360 degrees, divide it by the number of angles, it gives us each exterior angle of 24 degrees.

Or we could write 15 times 24 is 360 because that shows that if each exterior angle is 24 degrees and we have 15 of them, 15 lots of 24 make 360.

Either way, therefore it could be a regular polygon if all of the exterior angles are 24 degrees.

And question 5, Lucas draws a polygon, one of the exterior angles is 25 degrees.

Could it be regular and explain your reasons? Well, let's start with the fact again, that exterior angles in a polygon sum to 360 and write that down because it's the fact that we are working with.

And then 360 divided by 25 makes 14.

4, which is not an integer.

So 360 is not a multiple of 25 and a polygon cannot have 14.

4 sides, therefore the polygon cannot be regular.

So that is looking at exterior angles of regular polygons.

Now let's look at interior angles of regular polygons.

Whereas the sum of exterior angles is the same for any polygon.

The sum of interior angles in a polygon differs depending on the number of sides.

The sum of interior angles of a polygon can be found in multiple different ways, either by splitting it into triangles or by substituting the number of sides N into that expression or by any other method you know.

This means that the interior angles in a regular polygon could be calculated by using either the sum of the interior angles or the sum of the exterior angles.

Let's look at that together now.

Find the size of each interior angle in a regular pentagon.

Can you think how we might do this? Maybe pause the video at this point and have a little think about how we could approach this question before press and play to continue.

Let's solve this together now in two different ways.

For method one, let's consider the sum of the interior angles in the pentagon.

The sum of interior angles is equal to 180 multiplied by 3, which is what we get when we do 5, subtract 2.

That gives 540 degrees.

So we know those 5 angles must add up to 540 degrees.

And because it's regular, we know those 5 angles are equal to each other.

So if we do 540 divided by 5, we get 108 degrees for each interior angle.

Let's now do it again in a different way by considering the exterior angles.

The sum of exterior angles in any polygon is 360 degrees.

Therefore, if we do 360 divided by 5, we get that each exterior angle is 72 degrees and we also know that an exterior angle and it's adjacent interior angle, sum to 180 degrees.

So to get each interior angle, we can do 180, subtract 72 to get 108 degrees.

Same answer in two different ways.

Let's check what we've learned there.

What is the size of each interior angle in a hexagon? You've got three options A, B, and C.

Pause the video while you choose an answer and press play when you're ready to continue.

The answer is B, 120 degrees.

Find the size of each interior angle in an octagon.

Pause the video while you have a go at this and press play when you're ready for an answer.

The answer is 135 degrees.

And here are two different ways you can get that answer.

Let's apply this now in a different way.

The diagram shows a single vertex of a polygon, an interior angle is shown.

Determine whether it is possible for it to be regular.

Perhaps pause the video at this point and consider what approach you might take to this problem and then press play when you're ready to continue.

So let's try this now in a couple of different ways.

For our first method, let's use the formula for interior angles.

Here we have an equation where the left hand side of the equation is the expression that we can use to calculate the summit of interior angles when the number of sides is known.

That's 180 multiplied by N minus 2, where N is the number of sides.

Whereas on the right hand side of this equation we have 144, which is the size of that interior angle.

Now if it was regular, then that would mean that each of the interior angles would also be 144 and if we knew the number of sides, we could multiply 144 by that number of sides.

But as we don't know the number of sides, we can represent that calculation with the expression 144N.

So we've got an equation where N represents the number of sides and if we can rearrange and solve the equation, we'd see firstly whether or not N is an integer and if it is how many sides this shape would have.

Let's do that together.

We could expand the brackets to get 180N subtract 360 equals 144N and then we could add 360 to each side of this equation to get 180N equals 144N plus 360.

And then we could subtract 144N from each side of the equation to get 36N equals 360.

And then we to find the value of N, we could divide both sides of this equation by 36 to get N equals 10.

Now, the fact that N is an integer means that yes, it could be regular and it could be a regular polygon with 10 sides.

So let's conclude by saying therefore the shape could be a regular polygon with 10 sides, a decagon if all the other interior angles are also 144 degrees.

Let's look at another method now using the exterior angles.

We could first work out that if the interior angle at this vertex is 144, it would mean the exterior angle will be 36 degrees.

We can get that by doing 180, subtract 144 to get 36.

And then from here, we could just do the same thing again as what we were doing in the first learn cycle.

We could divide 360 degrees by 36 to work out the number of sides that gives 10.

And the fact it's an integer means that yes, it could be regular, therefore the same conclusion again, the shape could be a regular polygon with 10 sides, a decagon if all of the other interior angles are also 144 degrees.

Let's try this again now with a different interior angle.

The diagram shows a single vertex for polygon.

An interior angle is shown, determine whether it is possible for it to be regular.

Now you've seen two different methods for doing this.

Perhaps pause the video at this point and try this out with one of those two methods and see how you get on before pressing play to continue.

Right, let's try this first by using the sum of interior angles again and we'll set up a similar equation to last time.

If we expand the brackets, we get this.

If we add 360 to both sides, we get this and then subtract 125N from both sides to get this and then divide both sides of this equation by 55 and we get N equals 6.

545 and some more decimals.

The fact that it's not integer means that this could not be a regular polygon.

So let's conclude this working by writing the shape cannot be a regular polygon because N is not an integer.

Let's try this again now with the second method we had earlier using the sum of exterior angles.

If we know the interior angle here is 125, the exterior angle and its vertex will be 55.

We can get that by doing 180, subtract 125, let's get 55 and then we'll use the fact that the sum of exterior angles is 360 degrees.

So we divide 360 by 55 to get N equals 6.

45 and so on.

So once again, we can write the conclusion that the shape cannot be a regular polygon because N is not an integer.

Let's compare these two methods side by side.

Consider how they are different and how they are similar.

On the face of it, these methods appear quite different and they use different reasoning to get started, either the sum of the interior angles or the sum of exterior angles, but they both finish with the same two calculations.

If we look at these two sections of each method in method two, we did 180, subtract 125 to get 55.

And if we look at method one, can we see where that subtraction took place? To go from the top equation in that box to the next equation in that box, we subtracted 125N from both sides of the equation.

And to get the coefficient of N in that bottom equation, that box we did 180, subtract 125 to give us the coefficient of 55.

And then if we look at these sections of each method on method two, we did 360 divided by 55 to get 6.

545.

If we look at method one, can we see where we did the same calculation? To get from 55N equals 360 to the next line of that equation, we divided both sides by 55, which meant we did 360 divided by 55.

So while these methods look quite different and start in a different way, they both end in a very similar way.

So let's check what we've learned.

Can a regular polygon have interior angles of size 80 degrees? It's up to you how you work this out, but pause the video, work it out and press play when you're ready to work through it together.

Okay, if we work this out using exterior angles, we would do 180, subtract 80 to get an exterior angle of 100 and then if we divide 360 by 100 to see if all the exterior angles could be 100, we get 3.

6.

And in fact, it's not an integer means no, that polygon cannot be regular because 360 is not a multiple of 100.

Can a regular polygon have interior angles of size 162 degrees? Pause the video while you work it out, then press play when you're ready for an answer.

Once again, let's work through this using the exterior angles.

To get the exterior angle, we'll do 180, subtract 162 to get 18 as an exterior angle and then we'll divide 360 by 18 to get 20.

The fact it's an integer means that yes, it could be a regular 20 sided polygon, icosagon so long as all the other exterior angles are also 18 degrees.

Okay, it's over to you now for task B.

This task contains one question and here it is.

Here we have a table and each row of this table gives you some information about a polygon.

The top two rows say example one and example two.

You don't need to do anything with those two rows.

They just show you what's going on.

Example one is a polygon with interior angles of 140 degrees, exterior angles of 40 degrees.

Now the next column says, could it be regular? The answer is yes and there are a couple different ways you can work that out.

And it says then if yes, how many sides, 9 sides in this case.

But with example two, the interior angle is 145 degrees.

The exterior angle is 35 degrees.

When it says could it be regular, the answer's no.

And then the last column is left blank because the previous one said no.

And then for polygons, A, B, C, D and E, what you can see is either one or two of the boxes are filled in for you.

And what you need to do is work out what goes in all of the other blank boxes.

Pause the video while you have a go at this and then press play when you're ready to go through some answers.

Okay, well done with that.

Let's now work through some answers.

With polygon A, if the exterior angle is 65 degrees, then the interior angle is 115.

No, it can't be regular 'cause 65 is not going to 360.

Or you can work out of the interior angle and we're gonna leave that last column blank.

With polygon B, if the interior angle is 156 degrees, the exterior angle is 24.

And yes, it could be regular with 15 sides.

With polygon C, we know it's a regular polygon with 30 sides, so we just need to work out the interior and exterior angles, which are 168 degrees for the interior and 12 degrees for the exterior.

For polygon D, the interior angle is 150 degrees.

And then yes, it could be regular with 12 sides and we can justify that by showing how 12 times 30 is 360 or we could show about 360 divided by 30 is 12.

But polygon E, the interior angle is 30, the exterior angle is 150.

It cannot be regular and we can justify that by dividing 360 by 150 and we'll see it's not an integer answer.

So no, it's not regular.

You're doing great.

So let's now move on to the third and final learn cycle, which is finding missing angles with regular polygons.

So facts about interior and exterior angles could be used to solve missing angle problems involving adjacent regular polygons.

We have two polygons together in some sort of way.

For example, here we have a regular hexagon and a regular octagon and we can see that they share a common edge.

And it says find the value of X.

Now we're gonna work through this together shortly using two different methods, but perhaps pause the video at this point and consider what approach you might take before pressing play to continue.

Let's do this first, by using the interior angles.

We could work out that the sum of interior angles in a hexagon is 720 degrees.

Therefore, each interior angle in a hexagon is equal to 720 divided by 6, which is 120 degrees.

We can work out that the sum of interior angles in an octagon is 1080 degrees.

So to get each interior angle in the octagon, we could divide 1080 by 8 to get 135 degrees.

And then because those 3 angles we can see there are around a point and we know that angles around a point sum to 360 degrees.

We can do 360, subtract the sum of 120 and 135 to get 105 degrees.

Let's look at this again now, but using a different method using the exterior angles.

And the way we can do this is by drawing an extra line on this diagram that is an extension of that common edge and then working out the exterior angle on either side of that line.

We can work out each exterior angle on the hexagon by doing 360 divided by 6 to get 60 and we can work out each interior angle of the octagon by doing 360 divided by 8 to get 45.

And then we can see how the angle that was labelled X is the sum of 60 and 45 to get 105 degrees.

Same answer again.

Let's check how well we can do that ourselves.

Here we have a diagram with a regular pentagon and a regular hexagon with a common edge.

Find the value of X.

Pause the video while you work this out and press play when you're ready for an answer.

The answer is 132 degrees.

If you work this out by using the interior angles, your working might look something a bit like this.

And if you worked it out using the exterior angles, your working might look something a bit like this.

Facts about interior and exterior angles can also be used to solve missing angle problems involving nested regular polygons.

When we got one regular polygon inside another.

For example, here we have a regular hexagon with a regular pentagon nested inside it.

Again, they have a common edge, which is the base of those two shapes, but this time one shape is inside the other and it says, find the value of X.

Once again, we're gonna work through this using two different methods, but before we do, perhaps pause the video at this point and consider what approach you might take to it before pressing play to working through it together.

Let's do this first using the interior angles.

We can work out about each interior angle in the hexagon is 120 degrees.

Each interior angle in the pentagon is 108 degrees.

And then to get the value of X, well that's the difference between the 120 and the 108.

So we could do X equals 120, subtract 108 to get 12 degrees.

Alternatively, we could look at the exterior angles.

We could work out that each exterior angle in the pentagon is 72 degrees, each exterior angle in the hexagon is 60 degrees and then the value of X is the difference between those, which is 72, subtract 60 to get again 12 degrees.

Here's one for you to try.

We have a regular hexagon nested inside a regular octagon, and they share a common edge at the base.

You need to find the value of X.

Pause the video while you work through this and press play when you're ready for an answer.

Okay, the answer is 15 degrees and if you used the interior angles, your working might look something like this.

And if you used the exterior angles, your working might look something a bit like this.

Okay, it's over to you now for task C.

This task contains one question and here it is.

You are told by the question that all the shapes in the diagram are regular polygons and you need to find the value of each marked unknown angle.

So pause the video while you have a go at this and then press play when you're ready for some answers.

Okay, great job with that.

Let's now get some answers up.

The value of A is 90 or A degrees equals 90 degrees.

The value of B is 30.

The value of C is 24.

The value of D is 108 and the value of E is 132.

Let's summarise what we've learned in this lesson.

Exterior angles in a regular polygon are all equal and if we know that it means we can work out the size of each exterior angle by dividing 360 by the number of vertices.

We also know that the interior angles in a regular polygon are all equal.

So using that, we can work out the size of each interior angle in a regular polygon in a couple of different ways.

We can either work out the sum of the interior angles for that particular shape and then divide it by the number of vertices or we can do it by first working out the size of an exterior angle by dividing 360 by the number of vertices and then subtract our answer from 180 to get the interior angle.

And then also the number of sides of a regular polygon can be determined using either the interior angle or the exterior angle.