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Polygon Angle Sum - Concept

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

The sum of the angles in any polygon is equal to the number of sides in the polygon minus two, all multiplied by 180 degrees. Sum of polygon angles problems may ask you to determine the sum of angles in a particular type of polygon, the number of sides when given thhe sum of polygon angles, or a particular angle given the other angles in the polygon.

We need a formula that will tell us the sum of the angles in any polygon. So we're going to start by looking at a triangle, a square and pentagon. I've set up a table here where we're going to look at how many sides does it have, how many triangles can we fit inside that polygon and what's going to be the angle sum. So let's start with the triangle, well triangle has sides so I'm going to write 3 in for the number of sides, the number of triangles well this is just one triangle and we remember that the angle sum of a triangle is always 180 degrees. So let's move on to the square, well right now we don't have any triangles but if I drew in one diagonal I've now created 2 triangles. So the number of sides is 4, the number of triangles is 2 and if we have 2 triangles if I add that up that's going to be 360 degrees. So I'm starting to notice a pattern let's make sure it works for a pentagon. Pentagon we can draw in 2 diagonals from one vertex and now we've created 1, 2, 3 triangles. So we have 5 sides we've created 3 triangles and 3 times 180 is 540, so let's make this formula a little more general.
Let's say for an n sided polygon, the number of triangles that we're going to be able to draw is sum number. Well 3 minus 2 is 1, 4 minus 2 is 2, 5 minus 2 is 3, so you're taking the number of sides n and you're subtracting what number 2. So we said that will give us how many triangles we have, the number of triangles times 180 degrees because every triangle's angle sum is 180 degrees will equal the angle sum. So I'll write equals the angle sum, so notice we've created a formula.
If you want to know the sum of the angles in any polygon subtract 2 from however many sides you have and multiply by 180 degrees.

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