 ###### 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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# Equiangular Polygon Sums - Problem 2

Brian McCall ###### 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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Recall that an equiangular regular polygon is a polygon with congruent angles and sides. The formula for the total sum of angles in a polygon with n sides is (n - 2)*180°. The measure of an angle in an equiangular, regular polygon is the sum of the the angles divided by the number of sides. So, the formula for the measure of one angle is (n - 2)*180°/n.

From this formula, it is possible to find the number of sides given the measure of one angle of a regular polygon by solving for n.

In some problems, you’ll be given the measure of just one of those angles in our equiangular regular polygon and you’ll be asked to work backwards and figure out how many sides it has.

So let’s start off by writing what we know. We know that the measure of one of these angles is 150 degrees and that’s going to equal the number of sides minus 2 times 180 all divided by n. So if we solve this equation for n, then that will answer how many sides it has.

To do that I’m going to start by multiplying both sides by n which will undo dividing by n. So we have 150 times n, commutative property says I can switch them around and I’m going to distribute that 18. So we have 180 times n minus 360. So to solve this, I’m going to subtract 180n from both sides, minus 180n and I’m going to get a negative -30n equals -360, which is good. I want both of these to be negative or both of them to be positive, otherwise we’ll get a negative number of sides and that won’t make sense. So if we divide by -30, then we see that n equals 12. So how many sides does it have? It has 12, also known as a dodecagon.