 This video is one the interior angle of a regular polygon. We start out by talking about the definition of a regular polygon just to refresh a Regular polygon is a convex polygon where all sides are congruent and all angles are congruent This pentagon here would be considered a regular polygon if we were told That each of the five angles were congruent And each of the sides were also congruent then we could say yes, this is a regular polygon This next figure is also a pentagon because it has five sides But it is clear that each of the five angles are not congruent and the sides are not congruent So this would be a non regular pentagon our first question asks us to find the measure of one interior angle of a regular polygon We're starting with a pentagon again and if we remember we learned that the sum of interior angles is Given by the formula n minus 2 times 180 where we just have to find the number of sides of the polygon n So we'll start this by finding the sum of all of these angles with the formula that we learned previously We know that a pentagon has five sides So we're going to again plug that into our formula And when we get 5 minus 2 times 180 That gives us 540 degrees So we know that any pentagon when we measure the Sum of the angle the interior angles we are going to get 540 degrees But since we're told that this is a regular polygon We know that each of the five angles is going to be the same measure in this pentagon So if we take this 540 degrees That they have to add up to We know that that 540 has to be distributed equally between those five angles So if we want to find only one of those angles We can just take the sum of the angles 540 and divide by 5 540 divided by 5 is 108 and that means that each of these interior angles is 108 degrees and I don't need to write that all the way around But that's going to be our answer for one interior angle of a regular pentagon We can do the same thing for Finding one interior angle of a regular 15 gun first. We're going to Find the total angles with the n minus 2 times 180 If we know there's 15 sides and a 15 gun We're going to Plug that into our formula to find the total angles and we get 13 times 180 Which is 2340 degrees This represents the total angle interior angles of a regular or of a 15 gun We don't need to draw a 15 side figure to figure out What one interior angle is going to be all we'll have to do is? Take the total 2340 and again divide by how many sides there are 2340 divided by 15 Gives us 156 degrees So one interior angle of a 15 gun our answer is 156 degrees for each of those interior angles We can see a pattern here that for any regular polygon In order to find just one of the interior angles what we did for the previous problem and for this problem was to divide by The number of sides of that polygon So for our third it asks us to find the measure of one interior angle of an n-gon That's just asking us to fill in the formula for Finding an interior angle for any regular polygon no matter how many sides there are We know that it's going to be the total the sum of the interior angles divided by how many sides It's important to keep these straight, and that's why we have the table written out there So we need to look at the problem to see if they're asking us for the sum of the interior angles Or just one of the interior angles This will help us up the last problem in this video which asks us to find If we are given a regular polygon And we're given the measure of one interior angle of a hundred and sixty hundred and sixty two degrees We're asked to find how many sides are in that polygon So we're going to approach this with this formula that we just set up n minus two times 180 divided by n And I'm going to write that out here n minus two times 180 divided by n and remember that represents one interior angle Because we're dividing by the n This is a problem where they give us the measure of this one angle and We're asked to find how many sides we're trying to find out what n is So this is a problem where we're actually going to work backwards a little bit We'll fill in what we know we know one angle is a hundred and sixty two degrees And I'm going to set up the rest of this equation And now what we're trying to solve for in this equation is the n Keep in mind whenever we have a fraction in an equation We have a simple way to get rid of the fraction and make this an easier problem to work out If I want to simplify this equation I can multiply this side by n Which will cancel out the denominator and make this easier to solve But because I uh multiplied the right side by n I also have to even that out by multiplying the left side by n as well and what happens then Is I'm left with just n minus two times 180 on top And then when I multiply those two together I have a hundred and sixty two n This is simplified the equation now and now I can just distribute that 180 out and solve the rest of the problem I can actually distribute that 180 to both of those even though it looks a little bit backwards 180 times n is 180 n and 180 times negative two is negative 360 And so when I go ahead and Solve that I'm going to get all my n terms to one side Whoops. I didn't mean to Didn't mean to um Subtract by 162. I'm going to subtract by 180 n And I'm going to get on this side negative 18 n Equals that cancelled those out negative 360 In my last step then because I'm trying to get n by itself is to divide both sides by negative 18 And that negative over a negative Will cancel out and leave me with 360 divided by 18 Which is 20 n equals 20 This represents the number of sides and is the number of sides That will make this true We're in a regular polygon the measure of one interior angle is 162 degrees It's not a bad idea to just do a check on this I'll go over to the side here and go back plug in n equals 20 into my Formula for one interior angle That's going to give me hopefully my answer of 162 degrees And then I know that I indeed did that correctly If I do 18 times 180 divided by 20 I do get 162 so this gives me a check that n equals 20 is my answer for how many sides there are in the polygon