A regular nonagon is placed next to a different regular polygon as shown in the diagram. Calculate the number of sides that the regular polygon has.
So first of all, we can say that the nonagon is going to be our polygon on the left and the polygon on the right is our unknown polygon. So now, what we need to do is mark on any angles we know.
And we know that the interior angle of a regular nonagon is going to be 140 degrees. And the reason we know this is because we’ve worked it out. And the way we’ve worked it out is by using this formula.
And this formula tells us that the interior angle of a regular polygon, and this is very important we say regular polygon because what this means is that each of the angles are the same. And this is equal to 180 multiplied by 𝑛 minus two over 𝑛, where 𝑛 is the number of sides of our polygon.
And this formula actually comes from the fact that the 180 multiplied by 𝑛 minus two represents the smallest number of triangles that we can divide a polygon into. For example, if I’ve drawn here a hexagon and this is a regular hexagon, then if I draw a line from one point to the other points where all form a triangle — I’ve drawn three lines — and you can see that I formed four triangles.
And therefore, we can see we’d have 180 cause it’s 180 degrees the number of degrees in a triangle multiplied by and we’d have 𝑛 minus two. Well, it’ll be six minus two which will be four. And that’s because we’ve got four triangles as we’ve shown in the diagram.
Well, if you apply this to our nonagon, well a nonagon is a nine-sided shape. So we’d have 180 multiplied by nine minus two all divided by nine. And this will give us 1260. And that’s because 180 multiplied by seven is 1260 all divided by nine, which will be equal to 140 degrees which is what we’ve got here.
Well, now because we’ve got the 140 degrees, the 58 degrees, we can work out the interior angle of our unknown regular polygon, which I’ve called 𝑥. And the way we can do that is because we know that 𝑥 plus 140 plus 58 is gonna be equal to 360.
And the reason we’ve set up this equation is because we know that the angles around a point are equal to 360 degrees. As you can see here, I’ve given reasoning for my equation as it’s always important in an angle’s question to give reasoning whenever you’re working out an angle or making an equation.
So now, what we can do is we’re gonna actually look to solve this equation. So the first thing I’ve done is added 140 and 58. So we got 𝑥 plus 198 is equal to 360. So therefore, to find out what 𝑥 is, we need to subtract 198 from each side of the equation.
And we’ve done that because we want to leave 𝑥 on its own. So to do that, we take away 198 from 198 to leave us with zero. But whatever we do to one side of the equation, we must do to the other side of the equation. So we get 360 take away 198 which gives us 162.
So therefore, we can say that the interior angles of our regular unknown polygon are 162 degrees. But how is this gonna help us to solve the problem? Because what we want to do is calculate the number of sides that the regular polygon has.
Well, the first thing we can do to help us find the number of sides that the regular polygon has is to find the exterior angle. And we can do that because we know that the exterior angle is going to be 18 degrees. And the way we calculated that was to take 162 away from 180 which gave us 18. And we could do this because we know that the interior plus the exterior angle of a polygon is equal to 180 degrees.
Well, how is this gonna be important or useful? Well, it’s useful because we know that the exterior angle of a regular polygon is equal to 360 divided by the number of sides or 𝑛. So therefore, if we want to find 𝑛, so the number of sides, we can rearrange this formula.
And it’s gonna be 𝑛 is equal to 360 divided by the exterior angle, which will give us 360 divided by 18. And that’s cause 18 was our exterior angle. And this will give us a result of 20.
So therefore, we can say that the number of sides that the regular polygon has is 20 sides.