Lesson video

Hello, and welcome to another lesson on angles in polygons, regular interior, exterior angles, and the mean of irregular.

That's what we're going to be covering today.

A really, really amazing topic.

I know I say that so often, pretty much every video, but you starting getting used to it now.

The stuff we're doing is really powerful as usual.

So let's make sure those app notifications are silenced or that our phones are switched off, like mine currently is, and that we are ready to go, we're focused and we have got a silent room we can work in or as quiet as possible.

So without further a do, let's get on with Mr. T's lesson.

So for our try this, could we have a go at working out what the mean of each set of coloured angles are? Remember the mean, adding them all together, divide by the amount of numbers you have.

How could you describe each set of angles? Like, what are they? Are they like interior angles? Are they exterior angles? Are they something completely different? What do you notice and how would you apply it to other polygons? I'm going to give you 10 minutes to have a go at that now.

So pause the video and have a go, please.

Okay, let's go through try this then.

We've got the following then.

So find the mean of each set of coloured angles.

If I add together 135, 85, 130, 120 and 70.

I'll get the interior angles of a pentagon.

So that is going to give me, if I add all those together, What's it going to give me? Interior angles of a pentagon.

540 degrees, isn't it? And then dividing by five Should gives me 108 for those ones there.

So the blue angles sum to 108.

What about the pink ones? What does the pink one sum to? I've got 360 degrees when I add them all together and then divided by, how many have I got? Five altogether, an't I? That'll give me 72 degrees.

And that is the mean of the pinks.

So that is the pink.

That one of course is the blue here.

And then this one, I've got a one, two, three, four, five, six sided shape there, which is a? Hexagon, very good.

So that's a hexagon.

Now, if I want to work out the mean of those angles, if I add them all together, there we go, lovely.

If I add them all together, what do I get? I get, come on I'm not hearing you loud enough.

What is it? Once more? 720 degrees, very good.

So 720 degrees divided by one, two, three, four, five, six, isn't it? So if you divide that by six, what do you get? shout it, make sure that screen is absolutely just hearing every last moment of what you're saying.

120 degrees, right? What are you doing shouting at the screen for? Anyway, 120 degrees.

So when you do that, you get 120 degrees and that is our green ones, cool.

And then what about the yellow ones? That's going to be 360 again, isn't it? Because it's the exterior angles, isn't it? Good, so we've got one, two, three, four, five, six exterior angles.

So what's 360 divided by six? What is it? What is it? What is it? I'm still waiting for it, what is it? It's sixty, very good.

So 60 degrees is going to be our mean for the yellow ones.

Cool, how would you rather describe each set of angles? We've already been through that, but I've just said it verbally.

So that is the total interior, for the blue.

The pink is the exterior.

My apologies, the blue one would be the mean, mean interior, not the total, the mean.

The pink is the mean exterior.

The green, for this one is the mean interior, and the yellow is the mean exterior.

Apologies for this shocking handwriting there.

There you go.

It's Okay.

What do you notice and how would it apply to other polygons? That's what we're going to explore in just a moment.

So if you've already noticed it, keep it nice and quiet for now.

We're going to go on to that.

But if you can't think what that is just yet, don't fret, don't worry, we're going to go on and create a generalisation.

So this is what it is.

Is that we've got the mean interior angle in an n-sided polygon, is simply the total, I'm just going to sort of ring that round 'cause you've already done that.

That's the total.

We already know that.

And then we're just dividing it by the amount of sides that we have.

Which makes a lot of sense, doesn't it? That's the amount, goodness me, the lag on my computer, amount of sides.

So when we got that all combined together, that makes a lot of sense.

I add up all my numbers, I divide by the amount of numbers I've got.

I add up all my angles within n-sided polygon, I get 360, for example, in a square and then divided by the amount of sides I've got, four.

So what's that 360 divided by four? 90, I know that every angle in a square is 90 degrees 'cause it's got a right angle.

So I'm all good with that.

That kind of works.

That just approved it for us as an example for one of them.

Not approved necessarily, but I've used it as an example.

And then the mean exterior in an n-side the polygon, that's going to be 360 degrees divided by the amount of sides that we've got.

So again, that makes fairly intuitive sense.

I've got 360 degrees.

That's what they always sums to.

All those exterior angles sum to 360 degrees.

And then divided by the amount of sides I've got gives me, of course, the mean exterior angle.

So if I were to work it out for each of these, I could work it out, if I wanted to, but what's more powerful for these, we usually talk about regular polygons when we talk about the mean.

This is a regular pentagon.

That's a regular pentagon.

And what we can do here, is we can say, wow, what would it be? It would be 180, in a bracket n minus two.

So 180, five minus two.

180 times three.

If I do that, I get my total interior angle.

My total interior angle, therefore be 540 degrees.

And then if I divide that by the amount of angles, I'm distributing across one, two, three, four, five, that would give me 108.

So I know each of these angles is going to be 108 degrees, and I can fill that information in.

Now, some of you there are perhaps more switched on about this topic and are understanding this a bit more may think, yeah, cool, I can jump to a conclusion here.

I can say, what my exterior angle would be, continue that line on from there.

That would be what? 72 degrees, wouldn't it? Now we can check that now 'cause we've got this formula down here that made a lot of intuitive sense.

The total over the number of sides.

So with that, I can now check and I can do 360 divided by five.

And that gives me, what that angle? 72 degrees, goodness me.

So that makes a lot of sense.

I can then continue that line going there, and I get 72 degrees.

72 degrees, 72 degrees, and 72 degrees.

Remember that sort of closing image that we had going in there on approved exterior angles.

All right, very good.

So let's continue.

For your independent tasks, I'm going to give you a bit longer for this one, I'm going to give you 15 minutes to think about this 'cause some of it may be a little bit challenging.

So pause the video now and have a go at working out what they are.

So I'm going to go through the independent tasks now, the answers for it.

So the exterior and interior angle of a regular octagon.

If you're following my formula, I'm doing 180, n minus two over n.

So if I plugged that in, I'm going to get 180 into my calculator, times by, in a bracket, eight minus two, which gives me 1080.

And we should know that really to be fair of.

Total angles, we studied this for quite some time, and then divided by eight.

And that would give me 135 degrees.

And that's my interior, cool.

So what about the exterior then? You could do 180 subtract 135.

That's one way to get it.

So if I do that, that gives me? What does that give me? That gives me 45 degrees.

And that's my exterior.

But what I could also do is I could do 360 divided by eight.

And that gives me well known behold, that gives me 45 as well.

So that's another way that you could do if you wanted to.

Same idea for a decagon, but just we're doing it now with, we're doing 10 sides.

So we're going to do eight times 180, which is going to give us 1,440, divide that by 10 is going to give us 144 for our interior.

But what about the exterior? We can do 180 subtract 144.

And that gives us, what does that give us? Come on, I'm waiting, waiting, waiting, what is it? What is it? Say it louder, good.

36, so 36 degrees.

You're still shouting at a screen, what's wrong with you? We're doing 36 degrees for that one there.

This diagram shows part of a regular polygon.

How many sides does this polygon have? We know the exterior angle is equal to 20.

So I can say that 360 divided by the amount of sides is equal to 20.

Now I could multiply by n on both sides if I wanted to and I'd get, 360 would give us 20 n.

I can then divide it by 20 on both sides.

And then what do I get? I get 360 divided by 20 gives me? 18, doesn't it? So n is equal to 18 for that one.

So I know it's got 18 sides.

The sum of the interior angles of a polygon is 1,980.

Goodness me, quite a few.

Quite a few degrees in that polygon.

How many sides does that have? The interior angles are given as 180, n subtract two.

And that's going to be equal to 1,980.

So what I can do from there, is divide by 180 on both sides.

Now, if I do that, not expecting that you'll do that inside of your heads, But if I do that, I get 1,980 divided by 180, I get 11.

And that is equal to n subtract two.

Add two on both sides, I get n is equal to 13.

So I know that this has 13 sides, very good.

If you've got that, mark you work right or wrong.

If you didn't get all of it, then make sure you're absorbing in those mistakes and thinking, yeah cool, that's the reason I got that wrong Mr Thomas, lovely.

Let's continue.

So for your explore task, what I'd like you to think about is these two main questions here.

Find the regular polygon which has an interior angle of 156 degrees.

And then we've got two students who are squabbling.

They're sort of, they're fighting over, internally, within their brains, what's going on here.

And they can't really think of the polygon, that they're thinking of.

So you need to help them out and you need to say, this is the polygon.

This is how many sides it has got.

So pause the video now, I'll give you 10 minutes to have a go at that task.

Amazing, let's go through it then.

So we've got a lot of stuff we need to do, goodness me! Find the regular polygon which has an interior of 156 degrees.

If it's got an interior angle of 156 degrees, scratch my head for a moment thinking, ah, I can work out the exterior angle, can I? If I work out the exterior angles, it lies on that straight line and it's still 180 subtract 156.

And what does that give me? That gives me? What does it give me ? 24 degrees, right? So I know my exterior angle is equal to 24 degrees.

Now that's really helpful because if I'm now following this, I can do 360 divided by n is equal to the exterior, 24.

Times by n on both sides, times by n, I get 360 is equal to 24 n.

And then from there I can divide by 24 and I get n, of course, is equal to, 360 divided by 24 gives me 15.

So n is going to be equal to 15.

I know it's 15 sided polygon.

What about this one then? Again, this is very, very similar.

I'd actually say this is easier than the first question in some ways, 'cause you've been given the exterior angle is 15 degrees.

So the exterior in this case is equal to 15.

Again, follow the formula 360 divided by n is equal to 15.

So therefore, when I rearrange it, etc, I get n is going to be equal to? What's n going to be equal to? It's going to be equal to 360 divided by 15.

And what is it? Shout it out please.

24 degrees, very good.

So 24 sides.

What about this one? This is quite a special case.

This one is going to be 360 divided by n is equal to one.

And therefore, this is pretty cool, you can multiply by n on both sides and you'll get 360 is equal to n.

So it's a 360 sided polygon, how beautiful is that? Now, if you notice, this is a side point there.

I've been on about this before in this series going on about a megagon How cool is that? A megagon has a thousand sides.

It almost starts to look like, when you really, really draw that minute details, it tends towards looking like a circle.

I've drawn a really poorly constructed diagram over here, but you can see if you do tiny, tiny, tiny, little, sort of like, incremental sort of sides there, and there's this such large, there's such a large internal angles within it, they all tend towards that straight line, you'll eventually get to a circle which is, what's really amazing about this topic.

So it's with that, that we're at the end of the lesson now.

And I just want to say a big congratulations.

Of course, there are so many topics that come up there, both in terms of applied math and exam style sort of questions there.

So we've got loads of stuff we've covered today and loads of things that could potentially be on that exit quiz.

So make sure you absolutely smash it out the park, make sure you're doing the best job you possibly can and staying safe as well.

For now, take care and I shall see you soon, Bye, bye.