WEBVTT mathematics/geometry/pyo 00:00:00.000 --> 00:00:02.300 Welcome back to Educator.com. 00:00:02.300 --> 00:00:04.300 For the next lesson, we are going to go over polygons. 00:00:04.300 --> 00:00:12.400 We are going to talk about the different types of polygons and the interior and exterior angles of polygons. 00:00:12.400 --> 00:00:18.500 First, let's talk about what is a polygon and what is not a polygon. 00:00:18.500 --> 00:00:27.400 A polygon is a closed figure, formed by coplanar segments, such that the sides are non-collinear, 00:00:27.400 --> 00:00:33.600 and each side intersects exactly two other sides at their endpoints. 00:00:33.600 --> 00:00:55.300 Basically, a polygon ("poly" meaning many) is a closed shape (meaning it has to close), each side being a straight lines, and where no sides overlap. 00:00:55.300 --> 00:01:09.200 As long as we have a shape that is closed (nothing open--nothing can get through), with no overlapping, and... 00:01:09.200 --> 00:01:18.000 like this one...see how it is not straight...these are examples of polygons, and these are not. 00:01:18.000 --> 00:01:22.600 Now, it is OK if polygons look funny; if they look like this, that is OK. 00:01:22.600 --> 00:01:34.000 As long as it is a closed figure, each side is a straight line segment, and none of the sides overlap, then it is a polygon. 00:01:34.000 --> 00:01:40.800 So, here, because of that it is not a polygon; because of the overlap, it is not a polygon; 00:01:40.800 --> 00:01:48.900 and because this side right here is not straight, that is not a polygon. 00:01:48.900 --> 00:02:04.800 The two types of polygons are convex and concave; a convex polygon is when all of the sides are on the outside of the shape. 00:02:04.800 --> 00:02:11.800 What that means...maybe if I explain "concave," it will be easier to understand. 00:02:11.800 --> 00:02:20.400 A concave polygon is when two sides go in towards the center of the polygon. 00:02:20.400 --> 00:02:28.300 See how, right here, these two sides are angled towards the center; that would make this concave. 00:02:28.300 --> 00:02:33.600 The same thing happens here: we have this angle going towards the center. 00:02:33.600 --> 00:02:43.000 Think of a cave, like a mountain, or in the rocks; see how it goes inwards, and it creates a little cave? 00:02:43.000 --> 00:02:46.500 So, any time it does that, it is a concave polygon. 00:02:46.500 --> 00:02:55.100 If it doesn't, then it is a convex; so all of the angles are pointing away from the center. 00:02:55.100 --> 00:03:04.000 All of these angles are pointing away from the center, and that is convex. 00:03:04.000 --> 00:03:12.800 And this explanation here: No line containing a side of the polygon contains a point in the interior of the polygon. 00:03:12.800 --> 00:03:31.400 It just means that, if you were to draw each of these sides or extend them into lines, it is not going to cut through the polygon. 00:03:31.400 --> 00:03:36.300 If we make this into a line, it is not going to cut through the inside of the polygon. 00:03:36.300 --> 00:03:41.300 The same thing happens here, and the same thing here. 00:03:41.300 --> 00:03:55.400 With this one, however, if I draw a line, see how it cuts in the polygon; that is what it means--that is what this explanation is saying. 00:03:55.400 --> 00:04:06.300 The easiest way to remember: just think of the cave--it is creating a little space right there, like a cave, so they are concave polygons. 00:04:06.300 --> 00:04:21.800 Now, a regular polygon is a polygon with all sides congruent and all angles congruent; it is equilateral, and it is equiangular. 00:04:21.800 --> 00:04:28.800 That shows that it is equilateral, and this shows that it is equiangular. 00:04:28.800 --> 00:04:41.100 Any time that it is equilateral and equiangular, it is a regular polygon. 00:04:41.100 --> 00:04:50.200 And in order for it to be equilateral and equiangular, it has to be convex; you can't have a concave polygon that is equilateral and equiangular. 00:04:50.200 --> 00:04:54.300 Maybe it could be equilateral, but not equiangular. 00:04:54.300 --> 00:05:04.600 The interior angle sum theorem is to figure out the sum of all of the angles inside the polygon. 00:05:04.600 --> 00:05:18.400 If I have a triangle (which is actually going to be the first polygon that we are going to use), I have three angles in the triangle. 00:05:18.400 --> 00:05:28.700 The interior angle sum theorem is going to give me the total, or the sum, or the angle measure, of all three angles combined. 00:05:28.700 --> 00:05:37.200 So, if I have a quadrilateral (four angles), what do all four angles of the polygon add up to? 00:05:37.200 --> 00:05:51.100 First, let's start with triangles; a triangle has three sides...number of triangles: a triangle only has one triangle. 00:05:51.100 --> 00:05:55.700 We are going to talk about this in a little bit, but the number of triangles would just be one. 00:05:55.700 --> 00:06:07.800 The sum of angle measures: we know that all three angles of a triangle add up to 180. 00:06:07.800 --> 00:06:17.300 Next will be a quadrilateral, a four-sided polygon: number of sides: 4. 00:06:17.300 --> 00:06:33.700 Now, if I have a quadrilateral, I have two triangles; so the sum of the angle measures is going to be 360. 00:06:33.700 --> 00:06:40.000 I know that all of the angles added up together in a quadrilateral are going to add up to 360. 00:06:40.000 --> 00:07:06.800 And then, a pentagon is the next one: it has 5 sides; the number of triangles is going to be 3 (let's see if I can draw this: that would be 1, 2, 3). 00:07:06.800 --> 00:07:16.300 The sum of the angle measures: here, every time we add a triangle...every time we have one more side, 00:07:16.300 --> 00:07:22.200 it is like we are adding another triangle in the polygon, and then we add another 180, 00:07:22.200 --> 00:07:31.500 because, for every triangle that exists in the polygon, there is an additional 180. 00:07:31.500 --> 00:07:34.500 We always start with the triangle, because that is the first polygon. 00:07:34.500 --> 00:07:42.500 Then, when you get to quadrilateral, the next one, it is going to be plus 180. 00:07:42.500 --> 00:07:51.400 Then, to get to the next one, we are going to do + 180, which is going to be 540; 00:07:51.400 --> 00:07:59.800 so the angle sum of a pentagon is going to be 540 degrees. 00:07:59.800 --> 00:08:18.200 How about the next one, which is a hexagon?--6 sides: 1, 2, 3, 4... 00:08:18.200 --> 00:08:31.500 Again, we just add 180; it is going to be 720; and so on. 00:08:31.500 --> 00:08:36.200 For each triangle that exists, again, it is going to be 180 degrees. 00:08:36.200 --> 00:08:47.400 But what if I ask you for a 20-sided polygon--what is the interior angle sum of a 20-sided polygon? 00:08:47.400 --> 00:08:52.800 There is a formula to go with this, and that is right here. 00:08:52.800 --> 00:09:01.800 Because a triangle has 3 sides, but only one triangle exists; that is 180. 00:09:01.800 --> 00:09:15.400 For every additional triangle, it is going to be an additional 180; so here, isn't this 2 times 180?--because it is 180 + 180, which is 360. 00:09:15.400 --> 00:09:28.400 Here, from a 5-sided polygon, there are three triangles that exist, so isn't that 3 times 180 (180 + 180 + 180, which is 540)? 00:09:28.400 --> 00:09:36.800 So, it is 180 times the 3; here, there are four triangles, so I have to do 4 times 180. 00:09:36.800 --> 00:09:48.200 So, if I want to find a 20-sided polygon, how many triangles exist? 00:09:48.200 --> 00:10:01.200 Well, look at the pattern: 3:1, 4:2, 5:3, 6:4, 20...it is 2 less, so it is going to be 18. 00:10:01.200 --> 00:10:14.900 Now, again, this is to get 360 here; so we just do 2 times 180, which is going to equal 360; 3 times 180... 00:10:14.900 --> 00:10:21.600 the number of triangles times 180...4 times 180 was 720. 00:10:21.600 --> 00:10:28.800 So here, all I have to do is multiply 18 times 180. 00:10:28.800 --> 00:10:40.500 So first, I have to figure out how many sides I have; this is going to be n. 00:10:40.500 --> 00:10:49.600 And then, subtract the 2 to figure out how many triangles exist in that polygon; and then just multiply it by 180. 00:10:49.600 --> 00:11:24.200 So then, 180 times 18...0...this is 64...8 + 6 is 14...then put the 0 here; 0, 8, 1; 0, 4, 12...it is going to be 3240 degrees. 00:11:24.200 --> 00:11:35.600 All 20 angles of a 20-gon are going to add up to 3240 degrees. 00:11:35.600 --> 00:11:44.900 Looking at the formula, it is going to be the number of sides; subtract 2 (you are going to solve this out first) 00:11:44.900 --> 00:11:52.600 to figure out how many triangles you have in that polygon; and then just multiply it by 180; that is it. 00:11:52.600 --> 00:12:06.000 It is just the number of sides, minus 2: take that number and multiply it by 180, and that is going to give you the interior angle sum. 00:12:06.000 --> 00:12:21.900 Now, the exterior angle sum theorem ("exterior" meaning outside): whatever you have...it can only be one exterior angle 00:12:21.900 --> 00:12:37.100 from each side or vertex...then if this right here is 1, this is 2, this is 3, this is 4, and this is 5 00:12:37.100 --> 00:12:47.100 (there have to be 5 of them, because there are only 5 sides here; it is a pentagon)--all 5 angles here are going to add up to 360. 00:12:47.100 --> 00:12:50.700 And that is the exterior angle sum theorem; the interior angle sum theorem is different, 00:12:50.700 --> 00:12:59.300 because depending on the number of angles, depending on what the polygon is, the interior angle sum is going to be different. 00:12:59.300 --> 00:13:05.400 The more angles the polygon has, the greater the sum is going to be. 00:13:05.400 --> 00:13:14.900 But the exterior angle sum is always going to be 360--always, always, no matter what type of polygon you have, 00:13:14.900 --> 00:13:22.300 whether you have a triangle (if you have a triangle, it doesn't matter if you measure the exterior angle this way, 00:13:22.300 --> 00:13:40.800 as long as you do the same for each vertex: let's say 1, 2, 3 here; the measure of angle 1, plus...they are all going to add up to 360)... 00:13:40.800 --> 00:13:52.600 Here, the measure of angle 1 + 2 + 3 + 4 + 5 are all going to add up to 360; and that is the exterior angle sum theorem. 00:13:52.600 --> 00:13:58.100 The first example: Draw two figures that are polygons and two that are not. 00:13:58.100 --> 00:14:03.700 You can just draw any type of figures that you want. 00:14:03.700 --> 00:14:13.800 The first two that are polygons...you can just draw...it doesn't matter...any type of polygon. 00:14:13.800 --> 00:14:34.400 You can draw something like that, as long as it is closed, each side is a straight line segment, and no sides are overlapping. 00:14:34.400 --> 00:14:42.000 Then, two examples, two figures, that are not polygons would be exactly those things. 00:14:42.000 --> 00:14:55.100 Maybe like that...that would not be a polygon; if I have two sides crossing like that, that is a non-polygon. 00:14:55.100 --> 00:15:11.300 If I have, I don't know, something like this, that wouldn't be a polygon...use any examples that are something like this. 00:15:11.300 --> 00:15:16.000 These are polygons; these are not polygons. 00:15:16.000 --> 00:15:24.400 Moving on to the next example: Find the sum of the measures of the interior angles of each convex polygon. 00:15:24.400 --> 00:15:33.400 The first one is a heptagon: now, a heptagon is a 7-sided polygon; this has 7 sides. 00:15:33.400 --> 00:15:40.100 Remember: if the number of sides is 7 (n is 7), we have to figure out how many triangles. 00:15:40.100 --> 00:15:56.800 Remember: you subtract 2, so the number of triangles is going to be 5; and then, you are going to multiply that by 180. 00:15:56.800 --> 00:16:10.900 Now, the formula itself is going to be that the sum is equal to 180 times (n - 2). 00:16:10.900 --> 00:16:17.700 This is 7 - 2; that is 5; so it is the same thing as 180 times 5. 00:16:17.700 --> 00:16:35.400 Then, you can just do it on your calculator; I have a calculator here; 180 times 5 is going to be 900, so the sum is 900 for a heptagon. 00:16:35.400 --> 00:16:46.600 The next one is a 28-gon; now, once you pass 12-sided polygons, there is no name for it, so you would just write 28 with "-gon." 00:16:46.600 --> 00:17:01.600 This is a polygon with 28 sides, so n is 28; the number of triangles is 26; you are going to multiply that by 180. 00:17:01.600 --> 00:17:30.700 To use the formula, you are going to do 180 times number of sides; that is 28 - 2, so 180 times 26...use your calculator...is going to be 4680 degrees. 00:17:30.700 --> 00:17:41.500 So, all the interior angles of a 28-gon are going to add up to 4680 degrees. 00:17:41.500 --> 00:17:47.300 And the next one, x-gon: now, for this one, we don't know how many sides there are in this polygon. 00:17:47.300 --> 00:17:59.100 So, you are just going to use the formula; and so, we know that n is going to be x; the number of sides is x. 00:17:59.100 --> 00:18:02.800 In the formula, you are just going to replace the n with the x. 00:18:02.800 --> 00:18:08.200 It is supposed to be the number of sides, minus 2; instead, we are going to say x - 2. 00:18:08.200 --> 00:18:23.900 And that would be it; you are just replacing the n with whatever they give you as the number of sides, and that would be x. 00:18:23.900 --> 00:18:32.100 Given the measure of an exterior angle, find the number of sides of the polygon. 00:18:32.100 --> 00:18:39.800 Before we start with these numbers, with these examples here, I want to first use a triangle. 00:18:39.800 --> 00:18:47.400 Now, we know, from the exterior angle theorem, that the sum of the exterior angles is always 360. 00:18:47.400 --> 00:18:53.800 The sum of the measures of all of the exterior angles is going to be 360. 00:18:53.800 --> 00:19:05.800 If I have a triangle, here, here, and here: those are my three exterior angles: 1, 2, 3. 00:19:05.800 --> 00:19:12.800 How would I be able to find the number of sides? 00:19:12.800 --> 00:19:21.500 Well, in this case, how would I find each of these angle measures? 00:19:21.500 --> 00:19:32.500 Wouldn't I have to do 360 degrees, divided by the number of exterior angles? 00:19:32.500 --> 00:19:41.000 This is going to be what?--each of these angles has to be 120. 00:19:41.000 --> 00:19:47.700 Now, again, this is going to be for a regular polygon; for a regular polygon, this is going to be 120; 00:19:47.700 --> 00:19:54.000 if all of these exterior angles have the same measure, then it will be 120 each. 00:19:54.000 --> 00:19:57.100 So, that way, it will total to 360. 00:19:57.100 --> 00:20:05.100 Well, it is like they are giving you the measure of each exterior angle; so how would we figure it out... 00:20:05.100 --> 00:20:18.000 if I said that each exterior angle has an angle measure of 120--each exterior angle of a polygon is 120--find the number of sides. 00:20:18.000 --> 00:20:27.100 Well, you would have to do the same thing: 360 (because that is the total) divided by 120, and that is going to give you 3. 00:20:27.100 --> 00:20:32.800 So, you know that there are three sides here. 00:20:32.800 --> 00:20:40.700 The same thing works for this: 36 is the angle measure of each exterior angle. 00:20:40.700 --> 00:20:56.600 So, if 360 degrees is the sum of all of the exterior angles, divide it by 36 to find the number of sides; you get 10. 00:20:56.600 --> 00:21:06.600 That means that the polygon has 10 sides. 00:21:06.600 --> 00:21:33.000 The same thing works here: 360, divided by 15 degrees (you can use your calculator), is going to give you 24 sides. 00:21:33.000 --> 00:21:43.000 If there are 24 sides, each angle of the sides (because if there are 24 sides, that means that there are 24 angles)-- 00:21:43.000 --> 00:21:48.600 each exterior angle--has a measure of 15 degrees, which will then 00:21:48.600 --> 00:21:54.400 (since there are 24 of them) add up to 360 when you multiply these two together. 00:21:54.400 --> 00:22:07.800 The same thing works for this one, x: each exterior angle measure is going to be x degrees--we just have to divide it. 00:22:07.800 --> 00:22:16.700 And since we can't solve that out, this will just be the answer; you are just simplifying it out as much as possible, and that will be it. 00:22:16.700 --> 00:22:22.300 There is nothing else that you can do with that. 00:22:22.300 --> 00:22:27.800 The fourth example: Find the sum of the interior angles of each polygon. 00:22:27.800 --> 00:22:37.700 Now, notice how both of these polygons are not regular polygons; it doesn't look like it is equilateral; it doesn't look like it is equiangular. 00:22:37.700 --> 00:22:47.600 But it is OK, because we are just looking for the sum of all of the interior angles. 00:22:47.600 --> 00:22:54.300 Since it is not regular, we would not be able to find the measure of each angle. 00:22:54.300 --> 00:23:02.900 But instead, we can find the sum--what all of them add up to--because it depends on the polygon, not the type of angles inside the polygon. 00:23:02.900 --> 00:23:10.900 Here, we have 1, 2, 3, 4, 5, 6; so we have a hexagon, a 6-sided polygon. 00:23:10.900 --> 00:23:19.500 And that means, in a 6-sided polygon, that we have 4 triangles. 00:23:19.500 --> 00:23:29.300 Remember: we subtract 2; we get 4 triangles; and then we have to multiply this by 180. 00:23:29.300 --> 00:23:51.200 4 times 180...this is 32...720; that means that the sum of all of the angles inside here is going to be 720. 00:23:51.200 --> 00:24:02.900 Now, if, let's say (I am just going to add to this problem here), this was a regular polygon-- 00:24:02.900 --> 00:24:10.200 say that all of the sides are the same, and all of the angles are the same, so it is equilateral, and it is equiangular; 00:24:10.200 --> 00:24:13.000 and I want to find what the measure of each angle is, then. 00:24:13.000 --> 00:24:19.800 Since each of these angles are the same, and I know that all 6 angles together 00:24:19.800 --> 00:24:25.900 are going to add up to 720, how can I find the measure of just one of them? 00:24:25.900 --> 00:24:37.800 Since they are all the same, how can I find the measure of just this one, the measure of angle A, or the measure of 1? 00:24:37.800 --> 00:24:43.800 Since they all have the 720, and they are all the same--they all have the same measure, and there are 6 of them, 00:24:43.800 --> 00:24:59.100 I can just take 720 and divide it by 6; 720/6 is going to give me the measure of each of these angles. 00:24:59.100 --> 00:25:21.000 So then, here, you do 720 divided by 6; each of these angles is going to be 120; 120 here, 120 here, here, here, here, and here. 00:25:21.000 --> 00:25:27.600 And that is only if you have a regular polygon, meaning that all of the angles are the same. 00:25:27.600 --> 00:25:38.900 All of the angles have to be the same for you to be able to divide your angle sum to figure out each of these angle measures. 00:25:38.900 --> 00:25:44.700 The next one: here, this is to find the sum of all of the angles side. 00:25:44.700 --> 00:25:54.300 This is a quadrilateral; we only have four angles; so this is just going to be 180 times 2: let's just use the formula... 00:25:54.300 --> 00:26:09.000 180...n - 2 is the sum; 180...we have four sides, minus the 2, so that means we have two triangles; 00:26:09.000 --> 00:26:19.100 180 times 2, we know, is 360 (I said 360, and I wrote 320). 00:26:19.100 --> 00:26:33.400 Now, again, if all of these angles were the same, were congruent, this is equilateral and equiangular, so it is a regular polygon. 00:26:33.400 --> 00:26:46.200 Then, you would take 360; you can divide it by 4; and that would just be 90 degrees; that is if each of these angles were the same. 00:26:46.200 --> 00:26:53.800 Then, each of them would have a measure of 90; and we know that that would just make this a square, if it was an equilateral, equiangular quadrilateral. 00:26:53.800 --> 00:26:58.400 That would make it a square; then you would know that each of these angles would have to be a right angle. 00:26:58.400 --> 00:27:04.000 But for the sake of just knowing what to do if you have a polygon that is regular-- 00:27:04.000 --> 00:27:11.300 not just a quadrilateral, but any other type of regular polygon--you would just take the sum, 00:27:11.300 --> 00:27:21.100 and divide it by the number of angles you have. 00:27:21.100 --> 00:27:24.000 And that is it for this lesson; thank you for watching Educator.com.