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Sum of interior angles of a polygon

Video transcript

we already know that the sum of the interior angles of a triangle add up to 180 degrees so 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'm going to make it irregular just to show that whatever we do here probably it applies to any quadrilateral with four sides not just things that have right angles and parallel lines and all the rest actually let me that looks a little bit too close to being parallel so let me draw it like this 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 interior the measures of the interior angles of a triangle add up to 180 so maybe we can divide this into two triangles so we from this point right over here if we draw a line like this if we draw a line like this we've divided into two triangles and so if the measure of 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 that's going to be all 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 plus 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 C plus y and we already know a plus B plus C is 180 degrees 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 some into and then we just take 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 a general version we're just trying to figure out how many triangles can we fit into that thing so let me draw an irregular Pentagon so 1 2 3 4 5 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 that perfectly cover this Pentagon this is one triangle the other pending 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 when you take the sum of that one in 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 1 2 3 triangles so 3 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 just to get our first two triangles we have to use up 4 sides we have to use up all the four sides in this quadrilateral we have to use up 4 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 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 the hexagons 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 six sides and I can get one try Ingle 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 side so one out of that one and then one out of that one right over there so in general it seems like it seems like let's say so let's say that I have s sides s sides s sided polygon s sided polygon and I'll just assume we already saw the case for five sides or four sides five sides or six sides so we can assume that s is greater than four sides let's say I have an S sided polygon and I want to figure out how many triangles non-overlapping triangles that 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 the sum of the interior angles of that polygon so let's figure out the number of triangles as a function as a function of the number of sides so once again two of the sides or four of the sides are going to be used to make two triangles so this two sides right over there and then we have two sides right over there I can have I can draw one triangle over and I'm not even talk about what happens on the other the rest of the sides of the polygon you can imagine putting a big black piece of construction paper there might be other sides here I'm not even worried 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 two triangles and then no matter how many sides I have left over so I've already used for the sides but after that if I have all sorts of craziness here I could have all sorts of craziness here so let me draw it a little bit neater than that so I can have all sorts of craziness 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 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 ten is that right one two three four five six seven eight nine ten it is a decagon and I in this decagon four of the sides were used for two triangles so I got two triangles out of four of the sides and now the other six sides I was able to get a triangle each there's our six 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 ten so let me make sure that I count am I am I just not seeing something oh I see actually didn't I have to draw another line right over here these are two different sides it's tough 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 then 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 general generally think about it the first four sides the first four sides we're going to get two triangles we're going to get two triangles so let me write this down so our number of triangles number of triangles is going to be equal to two 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 four so the number of triangles are going to be two plus s minus four 2 plus s minus four is just s minus two so if I have an ends s sided polygon and if I have an S sided polygon I can get s minus two triangles that perfectly cover that polygon and they don't overlap with each other which tells us that an S side sided polygon if it has s minus two triangles that the interior angles in it are going to be s minus two times 180 degrees which is a pretty cool result so if someone told you that they had 102 sided polygon 102 sides so s is equal to 102 sides you can say okay 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 zeros behind it so it'd be 18,000 degrees for the interior angles of a hundred and two sided polygons