Lesson video

Hello, and welcome to this online lesson on angles in polygons, generalising angles in polygons part two.

So this is the second part of our sort of like mini sort of series that I'm compiling together on a generalising angles in polygons.

So as usual, please take a moment to clear away any potential distractions, be that your brother, your sister, your pet that you may have around, and find that quiet space where you're not going to be disturbed.

So make sure your app notifications are silenced, and that you have gotten rid of all those distractions, and we're all good to go.

Always, always, always, we're going to do very powerful maths, so make sure you're focusing as best you can, so you have the best chance of succeeding.

Without further ado, let's go ahead.

So if you'll try this, what I'd like you to do is I'd like you to think about Antoni's error.

So he's saying, "I split my shape into 6 triangles, so I know the total of the internal angles is 6 times 180 degrees, and that gives me 1080 degrees." What would you have done differently though? What'd you think? Can you calculate the sum with the internal angles of that hexagon? Hmm.

Let's have a look.

So pause the video now and have a go at that task for me please.

Okay, excellent.

Let's go through it then.

So what is Antoni's error? Well, he split it up into 6 triangles.

That's great.

We're happy with that.

But when we split it off, we need to think about the fact that, sure, we've got 180 degrees in each triangle, but has he done it from one specific point where they all intersect to this point here? Would you remember when we split triangles before, we've done it from here or any other sort of vertice, and what we've done is we've got this.

We've got 1, 2, 3, and 4 triangles.

So you can very clearly see there, I've got 1 triangle, 2 triangles, 3 triangles, 4 triangles.

So actually, he has made a slight mistake there, and that it should be 4 triangles.

So you've split off into 4 triangles now.

So 4 times 180.

What would that be? Give you a moment just to think about that.

4 times 180.

What'd you get? Shout it out.

I'm not hearing you loud enough.

Come on.

What is it? Shout it out for me.

720 degrees, right? Very good.

So it's 720 degrees as a result of doing that.

So he's made one very, very fundamental error.

It's that you cannot go from that centre point there, you need to go from one of the vertices, one of those corners there, in order to see how many triangles you need.

So without further ado, let's go into our connect task.

So I want to continue this a little bit further and explore a few other polygons that we have here.

Now, can you tell me what this polygon is here? It's got 1, 2, 3, 4, 5, 6, 7, 8 sides, so it's an? Octagon.

Very good.

Now, this is very special as well 'cause all the sides are the same length, therefore, it's a? Regular octagon, isn't it? Yeah? So I don't go from the centre point, do I? I always go from the vertice, don't I? So I go from here or I could go from here, here, here, et cetera, you get the idea.

I'm going to go from this one on in this case.

So what we do is we create triangles, don't we? So it's 1 triangle there, 2, 3, 4, 5, and 6.

1, 2, 3, 4, 5, and 6.

Looks like sort of like a beautiful shell, they look a sculpture of some sort.

Anyway.

Ignore my rambling.

So we've got 6 triangles there, and the internal angles in a triangle in total interior with a triangle are? 180 degrees, right? So we're going to do 6 multiplied by 180.

And what does that give us for the octagon? And octagons internal angles are going to be.

Shout out.

1080 degrees, good.

So I know that's my total interior angles in an octagon, 8 sides.

What about this shape here? What shape is that? Please, can you think about that one? It's got 1, 2, 3, 4, 5, 6 sides.

It's a hexagon.

Good.

So remember we go from the middle, right? Uh oh, no, we don't? We go from the vertice, so let's go from here.

So we're going to have 1 triangle, 2 triangles, 3 triangles, 4 triangles, 1, 2, 3, 4, one of those beautiful shells again.

You can tell I really like shells.

So we're going to do 4 times 180 degrees.

Now, 4 times 180.

What does that give us? Make sure I can hear it nice and loud.

What is it? Geez.

I can't hear you for goodness' sake.

You're shouting at your screen, I can't hear you, of course.

So you got, you got 720 degrees, don't you, right? I'm sure you said that though, right? So it's 720 degrees.

So that is my total interior angles in a.

What do we say? This was a six-sided shape? It was a hexagon, wasn't it? Very good.

So that's really interesting.

And even if we've got irregular shapes where they aren't all the same length all the way around, we just go from the vertice and we split it.

So let's move on.

What I'd like you to do for your independent task is I'd like you to do the exact same that I've just done previously.

So you may want to go back in the video and have a think about what we need to do.

So with that in mind, I'd like you to pause the video and have it go and work out what are the sum of their interior angles will be.

Fantastic.

Let's go through the answers then.

So what I'd do of course is I'd do the exact same thing here.

I'd do 1 shape, 1 triangle, 2 triangles, 3 triangles, 4 triangles, 1, 2, 3, 4.

And I'd get of course, 4 times 180, which gives me 720 degrees.

Very good.

Now what about this shape? Well, actually what we notice is it's exactly the same.

I'd just rotate slightly.

I've got 1, 2, 3, 4, so 4 times 180 again.

Tried to trick you there, don't know if you got that, but very good if you did.

So it's 720 degrees.

This one, though, if you were listening very, very carefully for that last one, that's an irregular polygon, isn't it? So I could go from any vertice I wanted to, really, but what makes a lot of sense? I could go to, let's go from this one here.

That's probably going to be the easiest to connect them all.

So if I do this one, I'd go 1 triangle, 2 triangles, 3 triangles going always to my vertice, 4 triangles, there yet.

So 1, 2, 3, 4, and that last one, 5.

So I've got to do 5 times 180 degrees, which gives me, of course, 5 times 180? 900, right? 900 degrees.

So that's my total interior angles for each of those.

I hope that makes some sort of sense there.

What I'd like you to do and for your explore task, the final thing we're going to do in this lesson, is I want you to basically culminate, give all that knowledge together, gather it up in every little, tiny little space in your brain.

And in relation to what we've learned, who's going to be right there? You've got so many of those different students saying 180 times N minus 360.

There'll be 2 less triangles than number of sides, so multiply that by 180.

N minus two in a bracket times by 180, et cetera, et cetera.

So what I'd like you to do is to think about that just for and moment and say, have that little internal argument in your brain.

So always have things in mathematician as a teacher, what's the best way to see this topic? What formula works best when we're teaching it? Those sorts of things.

So think about those statements and think, who's right? Pause the video now and have a go.

Excellent.

I'm going to assume you may need some help or that you are going through the answer now.

So here's some support if you need it.

Now, if I draw a triangle at the centre to each side.

Hmm.

What does that mean? If I draw a triangle from the centre to each side, it's 180 multiplied by the number of sides.

Okay.

And then subtract 360 degrees in the centre.

So that's really interesting.

If I drew a square then, we know what the total interior angles of a quadrilateral.

We should know that like that now, what is it? 360 degrees, right? So what I can do there is I can go to the certain centre, roughly, and then split it up.

So that's 1 triangle, 2 triangles, 3 triangles, 4 triangles, 1, 2, 3, 4.

Now, if I did that, what I'd get is I'd get 4 times 180, which would give me.

What would that give me? 720, right? So 720.

Now, of course, that's wrong, we know the total interior angles is 360, but then, we're told it's multiplied by the number of sides, cool, we've done that, then subtract 360 degrees in the centre.

So let's subtract 360 from that.

And what do we get? Well, hey Presto, we've got 360.

That's what we wanted.

Brilliant.

Big up the 360.

Very good.

So we, we know that that is going to be the total interior angles.

So actually that works.

That may have come to some surprise, right? Now, if we did 180 times by N minus 360, what would we get? Let's try with that example of the square.

If you did that and substituted N as 4, we get 180 times 4 and then subtract 360.

So what that gives you is 720 subtract 360, and bear in mind, we just did the calculation just a moment to go down there.

Then, don't think it takes a rocket scientist to work out.

We've got a 360 again.

So it's quite suspicious it's all working out here.

Well, what about this one up here? Well, what we've got here is 4 minus 2 in a bracket.

Remember BODMAS, we need to do that bit first.

So we do 2 times 180 and that gives us 360.

Gosh, this is interesting.

So we've got so many different ways to express that here and it's somehow working out.

What about this one here? There will be 2 less triangles than the number of sides.

Okay.

Makes sense.

So multiply that by 180.

So we have the number of sides is 4, 2 less than that gives us 2, so 2 times 180 gives us 360.

Now, you can explore this for yourself with a different a shape, not just a square, not just a four-sided quadrilateral, but you could explore it for a hexagon, an octagon, a decagon, all sorts of different ones if you wanted to, and you'd come to the same conclusion.

So important.

And all of those are different ways to express the total interior angles.

And you've just done like a mini investigation there into how it works.

That is so incredibly powerful.

If you can think about that on a broader scale, you can start to generalise so many things in both life and mathematics, becomes so powerful.

Honestly, if you've kept up with that or if you've thought about how to do that, and you had some vague idea, that's honestly a really, really great start, so well done.

So unbelievably, that brings us to the end of this episode you've been watching.

Some really, really, really important stuff that you've learned today, and you're able to absolutely smashed the exit quiz you're about to take.

So make sure you complete it to the best of your ability.

Until next time.

See you soon.

Bye bye.