The sum of the angles in a polygon is always equal to the number of sides in a polygon minus two, all multiplied by 180. Since the angles in an equiangular polygon are equal, the measure of one angle in any equiangular or regular polygon is simply the sum of polygon angles divided by the number of angles in the polygon. Knowing this information allows us to solve polygon problems with missing angle measurements.
Let's say I tell you, you have ten-sided figure it's eqiangular. What's going to be the measure of one of those angles.
To find that out we have to first back up. There's two key terms. equiangular and regular that are going to apply to what we're talking about here.
The first one is equiangular which means that all the angles in the polygon are congruent. It doesn't have to do anything with the sides. It just means the angles. You can remind yourself of that because it look like we have the word angle here.
Regular combines equilateral and equiangular. It says that all angles and all sides must be congruent to each other.
So we look at these two polygons right here. We have two pentagons, only one of these are equiangular. Well, here we have all sides marked as congruent. So this is not equiangular. If we look at this pentagon right here, all of your angles are marked as congruent to each other. So, yes, this one would be equiangular. If you put these together, you have a regular polygon, which is like this hexagon right here.
Notice that all the sides are congruent to each other, and all the angles are congruent to each other. But how do we calculate the measure of one of those angles? Well, to do that we need to look at our polygon angle sum. We said that the sum of the angles in a polygon is equal to N minus 2 times 180. To find that, we said how many triangles can we draw in a polygon. And that was always the number of sides minus 2. The sum of the angles in the triangle is 180.
So if you want to find just one angle in a regular polygon, you're going to take this formula, which is the quantity of N minus 2, times 180, and since all the angles are the same, you can just divide by the number of angles that you have.
So you're taking this formula and you're dividing by the number of sides that you have in your equiangular polygon or in your regular polygon.