Lesson video

Hello, and welcome to another lesson on angles in polygons.

Generalising and comparing generalisations is what we're going to do today.

So we're combining a lot of the knowledge we've learned so far and amalgamating into one by doing those generalisations and, exploring the matter a little bit more behind what we're doing, and comparing our generalisations to more specific context.

So, without further ado, remember, you need to make sure your phone is silenced and switched off, and make sure that you are doing the best you can do by, remember, being in that silent and quiet place so that we can focus on maths because we always do powerful maths in Mr. Thomas's lessons.

So, without further ado, let's take it away with Mr. T.

So, we'll try this, what I'd like you to do, is I'd like you to sort these statements based on, , these diagrams here.

So you've got some diagrams there that you're going to analyse, now you've got the pentagon and you've also got the hexagon there.

So you need to be able to place them, within those, those, Venn diagrams. So, pause the video now, and have a go at doing that task.

Awesome, let's go through the try this then.

So the sum of the purple angles is the same as the sum of the orange angles.

Well if we go forward, we can see that yes, they are vertically opposite aren't they? So vertically opposite angles are equal.

So we can say that's true, for certainly the pentagon and we can also say it's true for the hexagon as well, so it goes into our intersection.

Let's tick it off.

The mean of the purple angle is 60 degrees.

Well, we know it's a pentagon, over here, and we know the total interior angles of a pentagon, are going to be, what are they? We should know them pretty well by now, what are they? 540 degrees, right? So the total interior, is 540 degrees, the statement said that the mean of the pipeline was, 60 degrees, right? So we're going to divide it, by how many have we got there? We've got one, two, three, four, five, divided by five, and we get 108.

So that's our mean.

So unfortunately, that is not true, for the pentagon, but what about the hexagon? well the hexagon has a total interior angle that sum to 720.

So for that one, we then divide it by six, and if we do 720, divided by six, what would we get? 720 divided by six, we've done this one before, what is it? It's going to be, 120 degrees, right? So that's not true for that one there either.

So, that's not true for any of them so actually, that one, goes on the outsides.

All of the marked angles have different sizes.

Well, let's look at them.

All the marked angles have different sizes.

Well, it seems to be the case that yes, they are different sizes, aren't they? But actually, if we think about it, all of them, are the same here, we've got this one here the purple, is the same as that one there, same as that one there, because they're all vertically opposite aren't they? Vertically opposite, vertically opposite, et cetera.

So, that applies to both of the polygons as well.

So, all of the marked angles have different sizes.

Well, we can say, that, that one should go on the outside.

The sum of these purple angles is 540 degrees.

Well, we did the total interior angles for the pentagon, so applies just to our pentagon.

So we can say, that, we can put a D in there.

And then the sum of the blue angles is 360 degrees.

Well, that applies for this one here, one, two, three, four, five, and they're all exterior angles.

Now that's really important to know, because they're all extra angles tiers, can be true for this hexagon and the pentagon.

So we can put that, just that, and there is our try this completed.

Let's move on.

So, we want to compare some of those generalisations that we've seen so far.

So, we've heard, when we're talking about exterior angles, what have we heard so far? Can you ever think about what we've had so far? What have we had so far that refers to exteriors? Those exterior angles? They sum, to 360 degrees, don't they? And then the mean, is equal to 360, divided by n.

So, that's really helpful to know, cool.

What do we have for our interior? Well, for our interior, we've got 180 n minus two and we always talk about n remember, in terms of n is the number of sides, so that's something always really important to note throughout all of this.

So 180, n minus two in a bracket, so multiplied gives us the in total interior.

So these are both referring to our totals.

So both of those are our totals.

The mean then, is going to be equal to 180, n minus two in a bracket, over n.

So do you remember all that? Right? Very good if you can remember all that.

Now, when we go into more specific context, remember what we do, is just replace n there, don't we? We replace n here, we replace n there, and we know that it's just the sums to 360 for the exterior.

So actually in reality, there's nothing too tricky about all of this, everything we've done so far, it's just a case of remembering that formula, remembering those, what do we have to do for this bit here? We had to, draw out, one, two, three, four, five, six-sided shape, a hexagon, and then, we did, the total interior angles.

But the key idea of that, was that we went from one specific point in where this formed, those triangles.

So, we have six sides, and four triangles, and that's what the four triangles were referring to, right? That four of that.

So, I'm going to give you an independent task now, that's to do with all this.

And what I'd like you to do, is I'd like you to have a go at doing that independent task now for me, please.

So pause that video, and have a go at that independent task for 12 minutes.

Awesome, so let's go through the answers then.

So the sum of the interior angles of a regular hexagon is going to be, what we said, it was going to be, six minus two, which is four, times 180.

And what was that? Four times 180, 720 degrees, right? So that means that each interior angle is going to be 720 divided by, the amount of sides, six.

So if you do 720, divided by six, what would you get? Shout it out, 120 degrees, right? So 120 degrees goes there.

The sum of the exterior angles of a regular hexagon is? Well we should know that, that's definitely, that's the same throughout every single n-sided polygon, that'd be 360 degrees.

And then each exterior angle of a regular hexagon, is going to be 360 divided by n, which is going to be in this case, six.

So 360 divided by six, what does that give me? Well that gives me, shout it out, 60.

Very good, so 60 degrees for that one there.

The sum of the interior angles of an n-sided polygon is, well we had the formula for that just a moment ago.

So if you were paying attention, copying down those notes, this would be really nice and simple.

That would be, what would it be? 180, multiplied by n, subtract two in a bracket, lovely.

Each interior angle of an n-sided polygon is, taking that first bit, the sum, and then just simply dividing it by, n.

Very good.

The sum of the exterior angles, that one there, really nice.

360 degrees, always, always, always, 360 degrees.

And then each exterior angle of an n-sided polygon, 360, what was coming next? Divided by, and, get, so 360 divided by n.

Now for your explore task, I want you to substitute in those boxes, and fill them in with, what the answer will be.

So for example, I could take that first one and put, decagon, decagon, decagon, decagon, fill the rest with the decagon, and I get, all sorts of different answers.

So, all we're going to do, is very similar to what we did with our independent task, but then, we're going to generalise it.

And so, we're going to make specific examples and then we'll generalise it.

So pause the video now, and have a go at that task, for the next 15 minutes please.

Excellent, so let's go through the explore task then, so we've got the sum of the interior angles of a decagon is, well what is it? It's going to be, eight times, 180.

That would be of course, 1440 for a decagon.

Then if I repeat that throughout, decagon, decagon, decagon, each interior angle of a regular decagon, is going to be 140.

144, isn't it? Cause it's divided by 10.

The sum of the exterior angles is going to be 360, as it always is, and then each exterior angle will be 36 as a result.

You can always check your answer by, adding, both those together and they should give you, 180 degrees.

So that's our decagon done.

How about the next one then? So, let's go over an octagon this time.

So octagon, is an eight sided shape.

So if I were to do that one, then do it in a different colour, an octagon, the sum of the angles in an octagon, the interior angles, what would they be? That would be six times 180, wouldn't it be? So, six times 180, that's going to be, 1080 degrees.

Good, so then we can divide that by eight, and we get the interior angle, and that will be 135 degrees.

The sum of the exterior angles then therefore, would be, what would it be? What would it be? 360 of course.

And then this last one, each exterior angle would be 360 divided by eight, and that one there gives us, 45 degrees.

So we're halfway there, brilliant.

So what about the next bit? Well, next one we can take on is a nonagon, and I think the hardest bit of that, is realising, well, what is actually a nonagon? That is a nine-sided shape.

So it's got nine sides.

So with that in mind then, we can now think about the sum of the exterior angles is going to be seven times 180, which is going to be 1260.

So then each interior angle would therefore be that divided by nine, which can be 140 degrees.

So the sum of the exterior angles, I should really be proper and put, nonagon throughout all this.

But the sum of the exterior angles, would of course be, adding them all, together, with the sum of the exterior angles, and we get 360 degrees, don't we? So we can fill that one out to be 360 degrees, it's always going to be that.

And then each exterior angle therefore, is going to be 360 divided by nine, and that gives us 40.

We're almost there, we're on the last one now.

So the pentagon is five sides, now I've been through this quite a few times in this series, about pentagon.

So, the, sum of the interior angles of a pentagon, we've done that loads of times, that would of course be, 540 won't so? Degrees.

So then I can now think about, what would they, each of them be? What would each interior angle be? Well, 540 of course, divided by five.

We've done that a lot of times to so this can be 108.

The next one, would be 360, we know that, we know a lot of these at the back of our hand now, we should be really good with them.

And then the final one would be 72 degrees, cause we've done 360 divided by five.

Now the final one is a, generalisation of polygon and we've actually just done that, so as long as you were paying attention and focusing, for this bit, this should be really nice.

So we can just say, the sum of the interior angles in a polygon, this time would be what? What's the very general formula for this? What would it be? It would be 180, n minus two, wouldn't it? Where n, equals the number of sides.

Give yourselves a little smiley face for remembering that.

So each interior angle of an regular polygon is to be 180, n minus two, and then divided by n.

The sum of the exterior angles in a polygon, we've said that loads of times before, that would just simply be 360 degree, it doesn't matter what polygon we have, it will still be the same.

Then that final statement there, each exterior angle can be 360, divided by n.

So we've smashed it, we've done all of them now.

Very good, yes! So yet again, unbelievably, that takes us to the end of our lesson today.

I just want to say a big congratulations to you guys at home and girls, for doing such an amazing job there.

You've really smashed out of the park for you being able to keep up.

We're going to go on to our final lesson, next time, and we're going to be wrapping it all up and making it all come together, and we're going to be making sure that we really know polygons the best of our ability.

For now, take care, and I'll be seeing you.