First one is n minus 2 times 180. This equals the sum of the interior angles in a polygon. So all the interior angles added up. I just want to show you quickly where that comes from. So say for example you have a polygon like this.
Okay you can divide this into one, two, three triangles and we know the sum of the measures of the angles in a triangle add up to 180. This particular polygon I drew has one, two, three, four, five sides. Five minus two is three, and you can see there's three triangles there. So three times 180 equals 540. So that's where this formula comes into play, is you're going to have two less triangles, okay, than the number of sides or angles that you have in the polygon. So it's always going to be two less. Now this formula here, n minus two times 180 divided by n, we use this when you're trying to find the measure of one interior angle.
and a regular polygon. Now regular means that all the angles are the same and all the sides are the same length. So when you find out the total measure of all the interior angles and you divide that by how many angles you have, that'll give you the measure of just one of those interior angles. Okay this formula 360 divided by n is to find the measure of one exterior angle, that's the angle on the outside of the polygon. Again if it's a regular polygon, meaning that all the angles are the same, you take 360 divided by n and then this last one here is just that the sum of all the exterior angles in a polygon they're always going to add up to 360 it doesn't have to be regular they just they're always going to add up to 360 so for example here if i extend you know the sides like so these angles that are formed on the outside of the polygon here okay those are all going to add up to 360 like a circle okay so let's get into some examples i'll show you how these work Excuse me.
So in this polygon here, you can see we have one, two, three, four, five angles, or you could count the sides one, two, three, four, five sides. So we're going to do n minus two times 180 equals the sum of the angles. So we have five minus two times 180. which is 3 times 180, which is 540, okay? So if we want to solve for x, this missing angle here, we're going to add up all these angles, okay, plus x equals the total, 540. So let's do that.
So that's 120 plus 80 is 200, plus 100 is 300, plus this right angle here, that's 90, that's 390. So we have 390 plus x equals 540. So if we subtract 390, what do we get? We get 150 degrees. So that's this angle right here.
Now in this one here, excuse me, in this polygon here, we're trying to solve for the exterior angle. So what we're going to do here is we're going to add up all the angles, okay, and the exterior angles add up to 360. So in this one, we're just going to go over here and we're going to say, all right, 90 plus 90 plus 120, that's... 90 plus 90 plus 120 plus x, this angle here, equals 360. So that's 180 plus 120 is 300. Okay, so 300 plus x equals 360. And if we subtract 300, we can see that x equals 60 degrees. That's this angle right here.
Okay, now the next one, find the measure of an interior and an exterior angle of a decagon. And what I meant to write here is a regular decagon. So I'll just put that in there, a regular decagon, meaning all the angles and all the sides are the same.
So a decagon is like decade, right? There's 10 years in a decade. There's going to be 10 angles in a decagon.
So there's a couple different ways to do this. We can either use this formula here. Or we could use this formula here. I'm going to use this one here.
It's going to be a little bit easier since it's a regular decagon. So I'm just going to take 360 divided by 10. That equals 36 degrees. So each of the exterior angles of that decagon will be 36 degrees. But one thing that you'll notice is that, see when you extend this side, okay, like this, the interior angle and the exterior angle, they're going to be supplementary.
This might be a little bit easier to see here. So when I extended that side, the exterior angle and the interior angle, see how they form a line, a linear pair? They're going to add up to 180. So if this is 120, this is 60. If this is 36, right, the interior angle would be 180 minus 36, so that's 144. Okay, so sometimes it's easier to use this formula than this formula.
It's just a little bit simpler if it's a regular polygon, right? Okay, so then the last one. If the sum of the interior angles of a polygon add to 2,340, what type of polygon is it? Okay, so this one, it doesn't specify whether it's regular or not. It just says that all the interior angles add up to 2,340.
So we're going to go to this formula here, the sum of the interior angles. So it's helpful to memorize these formulas if you haven't already. And let's see if we can substitute what we have.
So we have 2,000. 340 equals n minus 2 times 180. Okay, I'm going to divide both sides by 180. Okay, this comes out to 13. And then if I add two to both sides, I get 15 equals n. So this is a 15-gon.
Okay, so when it starts to get up in those higher numbers, you can just put 15-gon, 15 sides or 15 angles-gon. So I hope you've been enjoying these videos. I hope you learned something. Sometimes it's just a little piece that's missing that can make a big difference in boosting your grade, and I hope these videos are helping you to do that.
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