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Polygon Angle Sum - Problem 2

Teacher/Instructor 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

Recall that the sum of angles in a polygon with n sides is (n - 2)*180°. So, given the sum of the angles, it is possible to find the number of sides by solving for n.

(n - 2)*180° = (sum of angles)

n - 2 = (sum of angles)/180°

n = (sum of angles)/180° + 2

Thus, given n sides, one can find the number of sides given the sum of the angles.

Let’s look at a problem where you have to work backwards from the angle formula. It says if the angle sum in a polygon is 1440 degrees, how many sides does it have? Again geometry teachers are not just happy that you can plug in numbers on one side, you have to be able to substitute it on both sides and solve for one of the variables.

So let’s first start with our formula. It says the angle sum in any polygon is equal to the number of sides minus 2, so this tells us how many triangles we can draw, times 180 degrees, because every triangle's sum of the angles is 180 degrees.

So what do we know? Do we know the angle sum or do we know the number of sides? Well it says how many sides does it have? So we’re solving for n. So I’m going to place angle sum with 1440 and we can solve this equation for n. Now it’s going to be linear because we’ve only got one n, we’re not multiplying by another one, so I’m going to distribute that 180 degrees. So we’ve got 1440 is equal 180 times n minus 360, because -2 times 180 is -360. So I’m going to add 360 to both sides here, we’re just doing simple algebra right now and we get 1800 is equal to 180 times m.

So I’m going to divide both sides by 180 and I’m going to come up here and I’m going to say that n, 180 goes into 1800 ten times, so how many sides does it have, it has ten, which is also known as a decagon.

The key thing here, start with your formula, figure out where you’re going to substitute and solve.

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