Angles with polygons
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Sum of interior angles of a polygon
We already know that the sum of the interior angles of a triangle add up to 180 degrees. So if the measure of this angle is a, the measure of this angle over here is b, and the measure of this angle is c, we know that a plus b plus c is equal to 180 degrees. But what happens when we have polygons with more than three sides? So let's try the case where we have a four-sided polygon-- a quadrilateral. And I am going to make it irregular just to show that whatever we do here it probably applies to any quadrilateral with four sides. Not just things that have right angles, and parallel lines, and all the rest. Actually, that looks a little bit too close to being parallel. So let me draw it like this. So the way you can think about it with a four sided quadrilateral, is well we already know about this-- the measures of the interior angles of a triangle add up to 180. So maybe we can divide this into two triangles. So from this point right over here, if we draw a line like this, we've divided it into two triangles. And so if the measure this angle is a, measure of this is b, measure of that is c, we know that a plus b plus c is equal to 180 degrees. And then if we call this over here x, this over here y, and that z, those are the measures of those angles. We know that x plus y plus z is equal to 180 degrees. And so if we want the measure of the sum of all of the interior angles, all of the interior angles are going to be b plus z-- that's two of the interior angles of this polygon-- plus this angle, which is just going to be a plus x. a plus x is that whole angle. The whole angle for the quadrilateral. Plus this whole angle, which is going to be c plus y. And we already know a plus b plus c is 180 degrees. And we know that z plus x plus y is equal to 180 degrees. So plus 180 degrees, which is equal to 360 degrees. So I think you see the general idea here. We just have to figure out how many triangles we can divide something into, and then we just multiply by 180 degrees since each of those triangles will have 180 degrees. Let's do one more particular example. And then we'll try to do a general version where we're just trying to figure out how many triangles can we fit into that thing. So let me draw an irregular pentagon. So one, two, three, four, five. So it looks like a little bit of a sideways house there. Once again, we can draw our triangles inside of this pentagon. So that would be one triangle there. That would be another triangle. So I'm able to draw three non-overlapping triangles that perfectly cover this pentagon. This is one triangle, the other triangle, and the other one. And we know each of those will have 180 degrees if we take the sum of their angles. And we also know that the sum of all of those interior angles are equal to the sum of the interior angles of the polygon as a whole. And to see that, clearly, this interior angle is one of the angles of the polygon. This is as well. But when you take the sum of this one and this one, then you're going to get that whole interior angle of the polygon. And when you take the sum of that one and that one, you get that entire one. And then when you take the sum of that one plus that one plus that one, you get that entire interior angle. So if you take the sum of all of the interior angles of all of these triangles, you're actually just finding the sum of all of the interior angles of the polygon. So in this case, you have one, two, three triangles. So three times 180 degrees is equal to what? 300 plus 240 is equal to 540 degrees. Now let's generalize it. And to generalize it, let's realize that just to get our first two triangles, we have to use up four sides. We have to use up all the four sides in this quadrilateral. We had to use up four of the five sides-- right here-- in this pentagon. One, two, and then three, four. So four sides give you two triangles. And it seems like, maybe, every incremental side you have after that, you can get another triangle out of it. Let's experiment with a hexagon. And I'm just going to try to see how many triangles I get out of it. So one, two, three, four, five, six sides. I get one triangle out of these two sides. One, two sides of the actual hexagon. I can get another triangle out of these two sides of the actual hexagon. And it looks like I can get another triangle out of each of the remaining sides. So one out of that one. And then one out of that one, right over there. So in general, it seems like-- let's say. So let's say that I have s sides. s-sided polygon. And I'll just assume-- we already saw the case for four sides, five sides, or six sides. So we can assume that s is greater than 4 sides. Let's say I have an s-sided polygon, and I want to figure out how many non-overlapping triangles will perfectly cover that polygon. How many can I fit inside of it? And then I just have to multiply the number of triangles times 180 degrees to figure out what are the sum of the interior angles of that polygon. So let's figure out the number of triangles as a function of the number of sides. So once again, four of the sides are going to be used to make two triangles. So those two sides right over there. And then we have two sides right over there. I can draw one triangle over-- and I'm not even going to talk about what happens on the rest of the sides of the polygon. You could imagine putting a big black piece of construction paper. There might be other sides here. I'm not going to even worry about them right now. So out of these two sides I can draw one triangle, just like that. Out of these two sides, I can draw another triangle right over there. So four sides used for two triangles. And then, no matter how many sides I have left over-- so I've already used four of the sides, but after that, if I have all sorts of craziness here. I could have all sorts of craziness here. Let me draw it a little bit neater than that. So I could have all sorts of craziness right over here. It looks like every other incremental side I can get another triangle out of it. So that's one triangle out of there, one triangle out of that side, one triangle out of that side, one triangle out of that side, and then one triangle out of this side. So for example, this figure that I've drawn is a very irregular-- one, two, three, four, five, six, seven, eight, nine, 10. Is that right? One, two, three, four, five, six, seven, eight, nine, 10. It is a decagon. And in this decagon, four of the sides were used for two triangles. So I got two triangles out of four of the sides. And out of the other six sides I was able to get a triangle each. These are six. This is one, two, three, four, five. Actually, let me make sure I'm counting the number of sides right. So I have one, two, three, four, five, six, seven, eight, nine, 10. So let me make sure. Did I count-- am I just not seeing something? Oh, I see. I actually didn't-- I have to draw another line right over here. These are two different sides, and so I have to draw another line right over here. I can get another triangle out of that right over there. And so there you have it. I have these two triangles out of four sides. And out of the other six remaining sides I get a triangle each. So plus six triangles. I got a total of eight triangles. And so we can generally think about it. The first four, sides we're going to get two triangles. So let me write this down. So our number of triangles is going to be equal to 2. And then, I've already used four sides. So the remaining sides I get a triangle each. So the remaining sides are going to be s minus 4. So the number of triangles are going to be 2 plus s minus 4. 2 plus s minus 4 is just s minus 2. So if I have an s-sided polygon, I can get s minus 2 triangles that perfectly cover that polygon and that don't overlap with each other, which tells us that an s-sided polygon, if it has s minus 2 triangles, that the interior angles in it are going to be s minus 2 times 180 degrees. Which is a pretty cool result. So if someone told you that they had a 102-sided polygon-- so s is equal to 102 sides. You can say, OK, the number of interior angles are going to be 102 minus 2. So it's going to be 100 times 180 degrees, which is equal to 180 with two more zeroes behind it. So it'd be 18,000 degrees for the interior angles of a 102-sided polygon.