Brian McCall

**Univ. of Wisconsin**

J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

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A **regular polygon** is a polygon in which all sides and angles are of equal length and measure. Recall that the formula for the sum of the measures of a polygon with n sides is (n - 2)*180°. To find the measure of one angle in a regular polygon, where all of the angles are the same, divide this formula by n, the number of sides. This would divide the total sum by the number of angles evenly between each angle.

For example, a regular nonagon has 9 sides. From the formula, the sum of the angles is (9 - 2)*180° = 7*180° = 1260°. Divided by 9, this is 1260°/9 = 140° so each angle in a regular nonagon has 140°.

Let’s look at a problem where you have to find the measure of just one angle in an Equiangular Polygon. It says what is the measure of one angle in a regular nonagon?

So two key terms first one is regular which tells you that all the sides and angles are congruent, and the other is nonagon something you have to have memorized. I can’t explain it to you enough is that you must have this memorized, so when you say nonagon, you must be thinking n equals 9. There are nine sides to this polygon. So what’s our formula?

Well we said that it’s going to be n minus 2 times 180 which will give us the sum of all the angles but then we have to divide that by n, the number of angles that are in our polygon. We said n equals 9, so what I’m, going to do is that I’m going to substitute in for n. So here we’re going to have 9 minus 2 times 180 all divided by 9.

So this is the measure of just one of those angles. Well we see 9 minus 2 is going to be 7, so we have 7 times 180 divided by 9 and now it’s okay to type this into your calculator and when you do that you should find that the measure of just one angle is 140 degrees.

So what did we do? We started with our polygon sum formula, we divided by the number of angles, and we substituted in how many sides we had.