Lesson video

Hi there.

Welcome to today's lesson.

My name is Mr. Peters.

And in this lesson today, we're gonna be thinking about how we can reason about the angles within different polygons.

This is a really interesting lesson and it really opened my eyes up when I spent more time thinking about how I can explain this to you.

If you're ready to get started, let's get going.

So by the end of this lesson today, you should be able to say that "I can reason about the sum of the angles within different polygons." This lesson today has got four keywords.

I'll have a go at saying them first and then you can repeat them after me.

The first word is angle, your turn.

The second word is sum, your turn.

The third word is vertex, your turn.

And finally the last word is polygon, your turn.

Let's have a think about what these words mean in a bit more detail.

Firstly, an angle is a measure of turn.

The sum is the result of adding two or more numbers.

A vertex is a point where two or more lines meet.

And finally, a polygon is a 2D shape made up of three or more straight lines.

This lesson today is broken down into two cycles.

The first cycle, we'll be thinking about angles and polygons.

And the second cycle, we'll be finding missing angles.

If you're ready to get started, let's get going.

Throughout this lesson today, we're gonna be joined by both Alex and Lucas.

They'll be sharing their thinking as well as any questions that they have to support us with our thinking and reasoning throughout the lesson.

So our lesson starts here.

Alex is looking at a number of triangles and he says he thinks he's noticed something.

He says that all of the angles in the triangle summed to 180 degrees.

And he knows this from some of his previous learning.

He's then gone to look at a number of quadrilaterals.

And he is also noticed that the angles in the quadrilateral sum to 360 degrees as well.

He says that each quadrilateral can be divided into two triangles when a line is drawn between two vertices.

Let's have a look.

There we go.

We can see that for example here.

So all quadrilaterals are also composed of two triangles.

Hmm, that's interesting, isn't it? We know that the sum of the angles in a triangle add to 180 degrees.

So we could say that each quadrilateral has two lots of 180 degrees.

There we go, which of course we know would sum to 360 degrees as the internal angles of a quadrilateral.

Hmm, great thinking, Alex.

I wonder how that works for other polygons as well.

Alex thinks pentagons can be composed of triangles as well and Lucas is agreeing.

Let's have a look.

There we go.

We can see, if we draw lines from the vertex to a different vertex within the shape itself, we can see now that we've created three triangles here.

And that is certainly only one way we could do this.

We could divide the shape up in lots of different ways, couldn't we? For example, like this.

So we can see that a pentagon could be divided into three triangles, couldn't it? Hmm, can you see what Alex is seeing? Quadrilaterals are composed of two triangles and pentagons can be composed from three triangles.

Hmm, he thinks there's a pattern emerging.

What do you think? Let's have a go at recording this in our table.

We can write the name of the polygon in the left-hand column and then we can write down in the middle column, the number of triangles that compose each polygon.

We can then go on to writing the sum of all of the angles in each polygon.

So, so far, we know that in a triangle, that obviously only makes one triangle and therefore that is one lot of 180 degrees.

Sum of the angles in a triangle is 180 degrees.

For a quadrilateral, we know that a quadrilateral can be composed of two triangles.

And as a result of that, we know that sum of the angles in a quadrilateral is two lots of 180 degrees, which is equal to 360 degrees.

And then what about the pentagon then? Well, we know that a pentagon when it's divided at the vertices, can be divided into three triangles.

That means that'd be three lots of 180 degrees, wouldn't it? And therefore, three multiplied by 180 would be equal to 540 degrees.

So we can say that the sum of the angles in the pentagon is equal to 540 degrees.

Hmm, that's really interesting.

And it might well be true, Lucas is saying, but he's still not quite convinced.

I wonder if we can try and convince him a little bit more.

Well, let's have a look then, shall we? Let's look at the composition of the pentagon here.

We can say that if the sum of the angles in a pentagon sum to 540 degrees, then we know that there are five internal angles within the pentagon itself.

And all of these angles are exactly the same, because it's a regular pentagon.

Therefore, we could divide the 540 degrees by the number of angles that we have, which in this case is five.

So 540 divided by five means that each angle has a value of 108 degrees.

Now if we look at each of the triangles here, we can see that all of these triangles here, are in fact isosceles triangles.

That's because they all have two lengths that are exactly the same.

And because they have two lengths that are exactly the same and they are isosceles triangles, we know that the opposite angles to the sides that are equal will also be the same.

So let's look at the triangle on the left-hand side to start off with.

If we subtract the known angle from 180, that would leave us with 72 degrees.

And of course, because it's an isosceles triangle, then we know that both the angles that are missing must be the same size.

So we can divide the 72 degrees, which is remaining by two.

Dividing 72 by two would be equal to 36 degrees.

So each of these angles here are equal to 36 degrees.

We know that this would also apply to the triangle on the right-hand side.

And then the triangle in the middle then.

Well, we can work out the size of the two bottom angles, can't we, by just working out the size of one of them.

We know that one of these angles is 108 degrees altogether.

However, we know part of the angle is 36 degrees.

So if we subtract the 36 degrees from the 108 degrees, that would leave us with 72 degrees.

So each of these angles at the bottom are 72 degrees, because it's an isosceles triangle.

And of course, to work out the angle at the top, we can then do 72 degrees plus 72 degrees, or two lots of 72 degrees and then subtract that from 180.

So that would be 180 minus 144, which would be equal to 36 degrees.

So the angle at the top is also equal to 36 degrees.

So we can say that the sum of the angles where the triangles meet together is always going to be equal to the angles at the vertex.

And in this case, that would be 108 degrees for the pentagon.

And it says, you can see that a lot more clearly in these examples here.

We've got three lots of 36 degrees at the top, which would be equal to 108 degrees.

And we can see this at the bottom as well where we've got a 72 degree and 36 degree angle, which again would be equal to 108 degrees.

We could also see from the top example that to work out the missing angle of the 36 degrees in the middle, we could have subtracted two lots of 36 degrees from the 108 degrees.

So we could have said 108 degrees minus 36 degrees, minus 36 degrees, which again left us with the 36 degrees, which was the size of the angle at the top of the middle isosceles triangle.

Lucas asked a really good question now.

He wonders, "Is this the case for all types of pentagons? Do the internal angles of an irregular pentagon also sum to 540 degrees?" Well, let's have a look.

Here's an example of an irregular pentagon.

We know a pentagon has five sides and an irregular pentagon means that the sides can be of different lengths.

Let's see if we can divide this up into three triangles as well.

There we go.

We can do that as well by drawing lines from some of the vertices.

So we can see here that we've made three triangles.

And I mean of course, we know that the angles in the triangle sum to 180 degrees.

So we can say we've got three lots of 180 degrees, which is also equal to 540 degrees, of course.

"So it does as well!" That's right.

It can work for either regular or irregular pentagons, can't it as well? And quite right, Lucas, what a great thing to start thinking about.

Maybe hexagons, does it work for them too? Okay, time for you to check your understanding now.

How many triangles can a hexagon be composed from where lines are drawn from only one vertex? Take a moment to think.

That's right.

It's C, isn't it? We can create four triangles within a hexagon.

Let's have a look.

There we go, there's one example about how you may have divided up the hexagon into four triangles.

And another quick check.

Can you calculate the sum of the internal angles of a hexagon? Take a moment to think.

Okay, so we know that there are four triangles altogether and the sum of the angles in each one of those triangles would be 180 degrees.

So we need four lots of 180 degrees and that would be equal to 720 degrees.

So the sum of the angles in a hexagon is equal to 720 degrees.

Okay, time for you to practise now.

What I'd like you to do, is continue our investigation looking at septagons, octagons, nonagons, decagons and even a dodecagon as well.

Just be aware of the number of sides that each one of these shapes has as you work through these.

As you're completing your investigation, what I'd like you to do is fill in the table as you go as well.

And here's just a small reminder that a dodecagon is a 12-sided shape and a hectagon is in fact a 100-sided shape.

I wonder how you'd be able to calculate that one.

Maybe ask yourself what you notice as you complete the table as you go along.

We'll help you to fill in the information for how many triangles and the sum of the angles in a hectagon now.

Good luck with that task and I'll see you back here shortly.

Okay, welcome back.

Should we have a look together then? So we can see here that each of these shapes now have been divided up into triangles leading from one vertex.

And as a result of that, we can see that a septagon can be divided into five triangles.

An octagon can be divided into six triangles.

A nonagon can be divided into seven triangles and a decagon can be divided into eight triangles.

What about a dodecagon? That's right.

A dodecagon could be divided into 10 triangles.

Ah, so we can see, we can multiply the number of triangles in each one of these shapes by 180 degrees each time, can't we? And so we can say that the sum of the angles in a septagon is 900 degrees.

The sum of the angles in an octagon is 1080 degrees.

The sum of the angles in a nonagon is 1,260 degrees.

The sum of the angles in a decagon is 1,440 degrees.

And finally, the sum of the angles in a dodecagon is in fact 1,800 degrees.

Now what did you notice as you were calculating these? Alex said that he noticed that it was always two less triangles that made up the shape, compared to the number of sides that each shape had.

So for example, a decagon had 10 sides, but it had eight triangles.

A septagon had seven sides, but in fact it was composed of five triangles.

So the number of triangles is always two less than the number of sides that a shape had.

So could we use this to deduce the angles of a hectagon then, I wonder? Well, we know that a hectagon is a 100-sided shape.

So that case then, it must be composed of 98 triangles.

And therefore we could do 98 multiplied by 180, which would be equal to 17,640 degrees.

So the total sum of the angles in a hectagon is 17,640 degrees.

That's a really interesting generalisation for us to take forward with our future learning.

Well done if you managed to get that for yourself too.

Okay and on to cycle two now, finding missing angles.

Let's start here, having a look at this shape here.

How could we find these missing angles, do you think? Well, Alex has got a good idea.

He's saying, "What information do we know already which could help us and we could jot this onto our diagram already?" Well, we can see that the unknown angles are a part of an isosceles triangle, aren't they? Because we have those two little lines, which are showing us that those side lengths are exactly the same.

And we know that we also have a regular pentagon here.

So that means all of the sides are the same length and all the angles will be the same as well.

So we can mark that on with these marks here to demonstrate that.

As Alex is saying, "We know that the sum of the angles in a hexagon is 540 degrees." Therefore, each one of the angles has a value of 108 degrees.

Because we can divide the sum of the angles by the number of angles that we have.

So 540 divided by five is equal to 108 degrees.

So now we know all of the angles inside the irregular hexagon.

And if we look carefully at angle a, we can see that forms part of an angle on a straight line.

We know that the angles on a straight line sum to 180 degrees.

And we know that part of the angle is 108 degrees, because it's from the hexagon.

Therefore, we can deduce that the missing angle a is 72 degrees.

Because we can subtract 108 degrees from 180 degrees, which leaves us with 72 degrees.

And of course we know it's an isosceles triangle.

And as a result of that, we know that angles that are opposite the equal sides are also equal to one another.

So the angle at the top of this triangle, would also be 72 degrees.

And therefore, to work out the size of angle b, we can add the two angles that we know and subtract that from 180.

'Cause we know that the angles in the triangle sum to 180 degrees.

So 180 degrees minus two lots of 72 degrees, which is 144 degrees.

So in total, that would be equal to 36 degrees.

So missing angle b is in fact 36 degrees.

Well done if you managed to work through that for yourself too.

What about this example here then? How could you work out missing angles c and d here? Lucas thinks he can solve this problem.

Let's see how he does it.

Again, he's gonna jot down the information that he knows already to help him.

So he knows that the perimeter of the shape has six sides, so it must be an irregular hexagon.

He knows that a hexagon can be divided into four triangles.

So the sum of the angles in a hexagon is equal to 720 degrees.

He knows that he can see from the shape that is made up of five right angles and one reflex angle.

Let's mark these on.

Here are five right angles, which all have a value of 90 degrees.

And here is our one reflex angle.

Because the bottom triangle is a right-angled triangle, we can find out the size of the missing angle next to angle c.

So we can say 90 degrees plus 28 degrees is equal to 118 degrees.

And then of course, subtract that from 180 degrees, which is the sum of the angles in the triangle.

That gives us 62 degrees.

So this angle here is 62 degrees.

We can now work out angle c, can't we? Because angle c is a part of the right angle, which is made up of the other 62 degrees.

So to do this, we know that a right angle is 90 degrees.

So if we subtract 62 degrees from this, that leaves us with 28 degrees.

So angle c is in fact 28 degrees.

I agree, nice work, Lucas.

You're doing some brilliant thinking so far.

And now to work out angle d then.

We know that this polygon has five right angles in it, which is five lots of 90 degrees altogether.

That's equal to 450 degrees.

And we know that the sum of the angles in a hexagon is equal to 720 degrees.

So we can find the size of the reflex angle by subtracting the 450 degrees from the 720 degrees.

720 degrees minus 450 degrees is equal to 270 degrees.

So we know that this reflex angle together is worth 270 degrees.

Hmm and now we can also then look at the second triangle up from the bottom and we can sum these angles together to find the size of the missing angle within that triangle.

So we can say that 18 degrees plus 28 degrees, plus the missing angle would be equal to 180 degrees.

Therefore, part of the reflex angle that is missing has a value of 134 degrees.

And of course, now that we know that part of the reflex angle is 134 degrees, we can find out the size of the other part of the reflex angle.

We can do 270 degrees minus 134 degrees, which gives us 136 degrees altogether.

Yeah, exactly that, Alex.

That really was nice work, Lucas.

Well done, you.

Okay, time to check our understanding now.

Can you find missing angle d here? Take a moment to have a think.

Okay, let's see how we could work this out.

Well, we know first of all it's an isosceles triangle within a rectangle, don't we? We know it's an isosceles triangle, because it's got those two little lines on either side, which show that those sides are the same length.

We also know that the opposite angles must be the same as well.

So we can start working this out.

We know that sum of the angles in the triangle is 180 degrees.

And therefore if we subtract 36 degrees from this, this gives us 144 degrees altogether.

Then we can divide the 144 degrees into two equal angles.

Because the angles opposite each other are the same.

So each one of these angles has a value of 72 degrees.

So d is 72 degrees.

Well done if you got that.

Hmm, what about this time then? Could you work out angle e this time? Well, because we know that the angles opposite the equal sides are the same, then we can say that this 72 degree angle is equal to this 72 degree angle here.

And now that we know that this angle is 72 degrees, that's obviously part of a right angle here formed by the rectangle around the outside.

So e plus 72 degrees is equal to 90 degrees.

So we can say that 90 degrees minus 72 degrees will be equal to e.

And as a result, e would be equal to 18 degrees.

Well done if you've got that too.

Okay and onto our final tasks for today then.

What I'd like to do is find the missing angles for each of these examples here.

And in order to do this, it might be useful to remember the internal angles of the different polygons that we've worked out so far.

Here's a list of the sum of the internal angles of the polygons that we've worked out so far.

Then I'd like you to extend it onto these problems here as well.

And once again, here's a list of the sum of the internal angles to help you along the way.

Good luck with those two tasks.

And I'll see you back here shortly to go through them.

Okay, welcome back.

Let's go through these together then.

Let's have a look at a then first of all.

We can see that it's an equilateral triangle within a square, can't we? So we know that the angles in the equilateral triangle are equal to 60 degrees.

So we can see that this angle here at the bottom would be equal to 60 degrees.

And then obviously, that forms part of a right angle in the bottom left-hand corner of the square.

So we can say 90 degrees minus 60 degrees, that would be equal to 30 degrees.

So angle a is equal to 30 degrees.

That angle b then, now that we know that angle a is 30 degrees, we can add that to the 90 degrees for the triangle on the left-hand side.

So we've got 90 degrees plus 30 degrees.

That's equal to 120 degrees.

Therefore, meaning that this angle here must be equal to 60 degrees.

We also know that it's an equilateral triangle.

So the angle at the top of the equilateral triangle, must also be 60 degrees.

And therefore, we've now got the angles on a straight line, haven't we? So we can now do 180 degrees minus 60 degrees, minus 60 degrees will leave us with the missing angle b, which is also of course equal to 60 degrees.

Well done if you've got that.

Okay, angle c now then.

Well, this here is a regular hexagon, isn't it? Now we know that the sum of the angles in a regular hexagon is equal to 720 degrees.

And if we divide that by the six equals sides, then that means that each angle is worth 120 degrees.

Because this is worth 120 degrees, then this angle is also worth 120 degrees.

And now we can use our knowledge of the understanding of the sum of the angles in a quadrilateral.

We've got 120 degrees plus 120 degrees plus 60 degrees.

And then all of that subtracted from 360 degrees, would be equal to angle c.

We've written that here as an equation.

We can see that c is equal to 360 degrees minus 240 degrees, which is the two 120 degree angles together.

And then minus the additional 60 degree angle.

That of course leaves us with another 60 degree angle.

So we can say that angle c is 60 degrees.

And now to attempt to work out angle d.

Well, we know that each angle in this hexagon has a value of 120 degrees.

If we already know that part of that angle is 60 degrees, then that means the remaining part is also 60 degrees.

This triangle that we can see is also part of a right angle, isn't it? So we can see that it's 90 degrees and 60 degrees added together, gives us 150 degrees.

And then if we minus all of that from 180, that would leave us with 30 degrees.

So angle d is worth 30 degrees.

And then finally, for e and f, this time, we've got an octagon.

We know that sum of the angles in an octagon add to 1080 degrees.

Therefore, we can work out each interior angle by dividing that by the number of angles that we have.

That would be 135 degrees.

Each angle here is worth 135 degrees.

Because we know that a trapezium is made up of two sets of the same angles, we can see here that angle e would be equivalent to the bottom corner of the other angle.

So at the moment, we've got two lots of 135 degrees and that's two angles of a quadrilateral, or in this case, the trapezium, which would be equal to 360 degrees.

So we can work out the size of these bottom angles by doing 360 degrees minus 270 degrees, which is the known angles that we have so far.

And then dividing those by two, that leaves us with the 45 degrees.

So angle e is also in fact 45 degrees.

And then finally for f, well, we know that the angle is 135 degrees altogether.

And we know a part of that angle, which would be 45 degrees.

So 135 degrees minus 45 degrees is equal to 90 degrees.

So f is equal to 90 degrees.

Well done if you've got all of those.

Okay and finally, the missing angles in these shapes here.

Well, to work out angle g, we can see it's an exterior angle, would be part of the angles in a full turn of 360 degrees if you took into account this angle and this angle here.

We know that both of these angles are 108 degrees, because they're both the interior angles of a regular pentagon.

So if both of these angles are worth 108 degrees, we can subtract both of those from 360 degrees, which gives us 144 degrees.

So g is equal to 144 degrees.

To work out h, we know it's an isosceles triangle.

This angle here is 108 degrees, because it's part of the regular pentagon.

And if we then subtract the 108 from 180 degrees, that would leave us with 72 degrees.

And of course if we divide that into two parts, each one of those angles would be 36 degrees, because it's an isosceles triangle.

And the angles opposite of the equal size are equal to one another.

Therefore, h is equal to 36 degrees.

Okay, let's work out angle i and angle j.

We know that here we've got an equilateral triangle.

And here we can see we've got a right angle.

Now we can work with the angles on a straight line.

We know that 60 and 90 is equal to 150.

So therefore, if we subtract both of those from 180, that means i would be equal to 30 degrees.

And then to work out j, we can use the angles on a straight line again to help us.

We know that part of that angle on a straight line is 60 degrees.

So the missing part would be equal to 120 degrees.

And then finally, for k and l, well, let's have a look at this shape.

We've in fact got octagon here.

We know that the sum of the angles within an octagon is 1080 degrees.

Therefore, each of the interior angles is worth 135 degrees.

'Cause we can divide the 1080 by the number of angles that there are altogether.

Therefore, if each internal angle is 135 degrees, then k is equal to 135 degrees of course.

And then to work out l, we can see we're working with a trapezium again here.

Therefore, we've got two sets of equal angles in a trapezium.

So 360 degrees minus 270 degrees is equal to 90 degrees.

And then we can divide that by two, which leaves us with 45 degrees.

So l is in fact 45 degrees, okay? And that's the end of our learning for today.

Hopefully you've enjoyed that.

And I'm feeling a lot more confident about reasoning about the angles in polygons.

To summarise what we've been thinking about, we can say that all regular and irregular polygons can be composed of triangles where lines have been drawn from one vertex.

You can find the sum of the interior angles of a polygon by multiplying the number of triangles that the polygon is composed of by the sum of the angles in a triangle, which is 180 degrees.

And finally, you can apply your knowledge for the sum of the angles in a polygon to help you find the missing angles.

That's the end of our learning for today.

I've really enjoyed that lesson, hopefully you did too.

Take care and I'll see you again soon.