How to Measure the Angles of a Polygon & Find the Sum

Regular Polygons

Only with regular polygons can you find the measurement of its angles. What makes a polygon, or a flat shape with straight lines, regular? A regular polygon is defined as a flat shape whose sides are all equal in length and whose angles are all equal. Visually, I like to think of these shapes as wannabe circles. These shapes so want to be circles that they are stretching their ends out as far as they can. Do you see this below, too?

These shapes are all regular polygons.

Do you see how if you drew a circle around them, each of the polygon’s corners would touch the circle? This is what makes a polygon a regular polygon. Because only regular polygons have a handy and useful formula for calculating the angles, these types of shapes are what we will be considering in this video lesson.

Sum of Interior Angles

The first angle measurement we will discuss is the sum of the measure of interior angles. Long name, I know. All it means is that we are going to find the total measurement of all the interior angles combined. What are the interior angles, you ask? The interior angles are the angles you see inside the polygon at every corner. So a triangle, for example, has three interior angles because it has three corners. A pentagon has five interior angles because it has five corners. Do you see how it works now?

It’s a good thing for us that we have a very useful formula we can use to find this sum.

Sum of the Measure of Interior Angles = (n – 2) * 180

Yes, the formula tells us to subtract 2 from n, which is the total number of sides the polygon has, and then to multiply that by 180. We can check this formula to see if it works out. We know that the angles of a triangle will always add up to 180. So, let’s try finding the sum of interior angles of a regular triangle. We have three sides, so our n is 3. We plug that in, and we get (3 – 2) * 180 = 1 * 180 = 180. A-ha! It works!

So, whatever regular polygon you have, to find the sum of the measure of interior angles, all you have to do is plug in your number of sides into the n variable and then evaluate.

One Interior Angle

Now, what about the measure of just one of the interior angles? What do you think we would need to do to find the measure of just one angle if we already know how to find the sum of their measures and that for a regular polygon all the angles are equal? Yes, we would need to divide our sum by the number of sides we have. And that is what our formula tells us to do.

Measure of Interior Angle = (n – 2) * 180 / n

Let’s go back to our triangle. We know that an equilateral triangle, a regular triangle, has all of its angles measuring 60 degrees. So, does this check out with our formula? Let’s find out. Our regular polygon is a triangle with 3 sides, so our n equals 3. We plug that into our equation, and we get (3 – 2) * 180 / 3 = 1 * 180 / 3 = 180 / 3 = 60. Look at that! It works, too!

Likewise, with the sum of the measure of interior angles, we can use this to find the measure of any interior angle of any regular polygon. All we need to do is to plug in our number of sides into the equation.