The total sum of the interior angles of two triangles would be equal to...

If I have a regular pentagon, how many triangles from one distinct point internally can I create?

If I have a regular pentagon, what would the total interior angles sum to?

If I have a regular octagon, how many triangles from one distinct point internally can I create?

If I have a regular octagon, what would the total interior angles sum to?

Polygons and triangles

﻿Hello, and welcome to this online video lesson on angles in polygons, generalising angles in polygons part one. So this is part of a two parts lesson that we're going to be exploring where we're going to start to generalise the amount of angles that are in polygon side the total interior, sum of the things they're involving triangles, et cetera. So building on everything we've learned so far, so without further ado we'll get started, but just before we get started, remember, we need to make sure that we've got our phones on silent if we are using our phones. Make sure those app notifications are turned off. And we're also going to make sure we're going to find a quiet spot in order to concentrate as best we can. Always like always, we will be doing some really interesting math, so let's make sure we're focused and we can do this to the best of our ability. Let's go on then. So, I'd like you to have a go at the, try this. Now we have here some various shapes that we split up into triangles. So what I'd like you to do is to give the name of them. I'd like you to tell me the number of sides, the number of triangles, and therefore the sum of the interior angles. So pause the video now, and have a go at that exercise. Fantastic. I assume that you've done this and we're going to go through it now. So the first one, we call that a quadrilateral, don't we? Because it's got four sides. So quad ri-lateral. So that's what we call a quadrilateral it has four sides of course, one, two, three, four. Number of triangles, we can very clearly see that it's two. And the sum of interior, you probably know that off the top of your head. You don't usually think about, it's 360 degrees, right? So what about this one? This one's a bit stranger. This is a Pentagon, it's got five sides, of course. Number of triangles, very clearly see that's going to be, three of course. We've encountered it when I've, when I've been teaching you, we've been encountering this so far, quite a lot that sum of the interior angles of a pentagon, hopefully you should be able to recall them fairly well now. So that one there, of course it's going to be 540 degrees. Right? Very good. So that one there is 540 degrees. Beautiful. Let's move on to this one. Then this is slightly strange. This has got one, two, three, four, five, six sides. So maybe it's easy to fill in the number of sides first cause we know it's six sides. So that's going to be, what was that called? Say, hexagon, right? If you want to be really specific, bonus point for saying it's an irregular hexagon, right? You could also add that in there, that it's an irregular. Irregular, there we go, good. So in a irregular hexagon, this is also an irregular pentagon. I'm just going to mark on with an I, and then you could actually also say this is like an irregular quadrilateral if you wanted to. Use less so here, of course. So it has six sides. We've also got one, two, three, four triangles. Now the sum of the interior angles, well, this was 540, we had three triangles. So this was going to be four times 180. And again, we've done that quite a lot so far in this sort of video lesson. So tutorial sort of series so far. So that one there is going to be 720 degrees. Hopefully you should be able to recall them fairly well. Now those small ones, right? And then finally we've got one, two, three, four, five, six, seven, seven sides. Right? Now I'm going to come back to the name just a moment, 'cause I'm going to let you think about that. But the number of triangles in that, well, how many triangles have we got? We've got one, two, three, four, five. Five triangles. Now, when we thought about this before, what have we done? We've seen that the number of triangles there is what? Number of triangles timed by 180, right? So what, what do I need to do? Well, five times 180 gives me, what is that give you? Five and 180. 900, right? So 900 degrees. So that's the total sum of the interior angles now seven sided shape, you may not know this, but that is actually called a heptagon. Heptagon. Not many people know that. So, if you actually got that, then very good, well done. Let's move on then. So we're going to think about what we've just done there just a moment ago, we're going to start to sort of like reject that altogether and really formalise it now. So number of sides in a triangle and the number of internal triangle, well of course, it's just going to be one. So some of the interior angles of course we know that it's going to be 180 degrees. Now the quadrilateral, we can say almost instinctively now very, very quick, we can tell that it's going to be 360 degrees. The pentagon, where we had that just a moment ago, what was that? 540 degrees, right? 540 degrees. Now a hexagon, we had that again, 720 degrees, wasn't it? Now a decagon, that's a strange one, that's a 10 sided shape. Number of internal triangles there is eight. So you're going to have to times that by 180 and eight times a 180, try not to use a calculator, eight times 180. What would that give you? Give you a moment just to be able to think about that. That was going to give you 1,440 degrees, right? So 1,440 degrees. Now the dodecagon is a 12 sided shape. And that has 10 internal triangles will times that by 180 degrees and we get of course that was nice and easy, 1,800 degrees. Now this is what's really special is we can now consider an N sided shape in an N-gon, right? So number of sides being N, now the total, the number of internal triangles here, if you notice we've just subtracted two each time, haven't we? So try to subtract two, et cetera. So if I wanted to evolve with expression evolving N then I'd need to subtract two from it, I just do N subtract two. Now I need to multiply that by 180. So what I'm left with is because I want to multiply the whole thing there by 180. What I'm left with is 180 multiplied by N subtract two. And that's a really powerful formula just to show you how powerful it is, I'm going to sort of wiggle it all around in like a bubble, right? That is a really, really powerful formula that we need to know going forward. Right? So, so, so, so, so imperatively important we know that. So that's going to be a really helpful formula now. So there's sum of the interior angles in an N sided polygon, is this okay? Lovely, let's move on. So here's your independent tasks. I've got some really, really interesting polygons for you to analyse here. And there are some that have got a thousand sides, right? You need to look up any way that you want, but look up how many, what that's called, how many internal triangles that may have, think about that. Using the stuff that we just learned in the connect task. So think about what the sum of the interior angles all those will be as well. So have a go at that task, the best of your ability, like I said, you may need to research quite a lot of this, okay. Best of luck. Pause the video, have a go at that now. Fantastic, let's go on then. So we've got our answers just here. Now this you may have had to see by looking it up, it's called a megagon, it's one of my favourite shapes. The number of sides being a thousand, that's almost like a circle when you've drawn that, right? So what you notice is these all tend towards a circle. When you draw a thing with loads and loads of sides, it eventually starts to look a little bit like a circle poorly drawn, but you get the idea, right? It's very, very close to looking like a circle. The number of internal triangles of course is just you subtract two from it, like we did just a moment ago. So subtract two, subtract two, et cetera. Not for that one there, my apologies for this one down here. We times that of course by, what'd we times the number of internal triangles by? 180, of course. So times that by 180 instead, and we get these answers just here. So hopefully by the time you realised, oh, okay, these are the amount of sides Mr. Thomas, cool, I get this. We can then see, well, the number of internal triangles subtract two, going back from what we learned previously, and then times by 180. So nothing too tricky once you realise that subtlety, that little small thing. So, we've got our Explorer task now, what I'd like you to do is I think you can do this without too much support. So I'm going to leave you guys to have a go at this guys and girls, to have a go at this task and analyse what's being said there, and just think about it for a moment. So this is one of those moments where I'm actually not going to provide support because I think, you know what you can do here. I have, I have faith I've confidence in the fact that you can do this. Read that statement and see what you can spot. Pause video now, have a go. Okay. Let's have a think about that statement then, so ACD combined is a pentagon, the sum of the interior angles of course is going to be 540 degrees. Well, if we go through ACD, well, that's going to look like this. So I've just outlined there with my red marker, what that looks like. And of course it's a one, two, three, four, five. So it is a five sided shape, so it is a pentagon, we know that. Some of the interior angles is 540 degrees, well, what was our formula? It was N minus two times by 118, that gives you three times by 180, which is 540 degrees. So that's correct, right? We know that statement is correct. What other shows? I'm sure you've found some others. If I just chose, for example, let's just do it in a jacket line this time, if I chose B C, right? So if I chose BC that'd be, one, two, three, four sided shape. So it'd be four minus two times 182, well I think without me having to even do some of these calculations, you may have realised four sided shapes going to be 360 degrees, of course, because you're doing two times 180, and that gives you 360. So, I'm sure you found so many more. I mean, there is an, there's a real, vast expanse of how many you can do. If I had to cover all of them, I'd be here well and truly until the next day I imagined. But, I'm sure you can use your common sense there to realise, oh, I could have D and E and that'll give me a you know, whatever sided shape and go from there. So I trust you can get on with that with yourselves and explore that further, because it is a very interesting task for this, so many possibilities, I wouldn't want to exhaust both ourselves there. So that brings us to the end of the lesson. Now I'm sure you've done some really good stuff there and learning that that is an, you know, for an N sided shape that you need to multiply by and subtract two and then multiply the whole thing by 180. That is incredibly powerful. And we're going to use that again going forward. So really, really make sure you are taking that on board. Remember, don't forget to do that exit quiz so that you can smash that learning as much as possible and prove just how much you've learned. Mr. Thomas really wants to see how much you've learned so important that you learn lots. And as always, I hope you stay safe. I'll be seeing you in the next episode. Take care for now. Bye bye.

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Generalising angles in polygons (Part I)

In this lesson, we will learn how to generalise the sum of the interior angles in an n-sided polygon.

Maths

Y8 Maths

Key Stage 3

Year 8

Generalising angles in polygons (Part 1)

Angles in polygons

Geometry

THEME

UNIT

Interior angles in a triangle

Categorising and defining polygons

Building shapes from triangles (Part 1)

Building shapes from triangles (Part 2)

Generalising angles in polygons (Part 2)

Finding missing angles in polygons

Exterior angles

Regular interior and exterior angles (and mean of irregular)

Generalising and comparing generalisations

Angle notation and problem solving

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Lesson overview:Generalising angles in polygons (Part 1)

Core Content

Angles in polygons: