Why does the formula 180(n 2) work

Let's consider what happens when you draw these triangles. They all come out from one point in the center, n in total. We know that can't be the right value for the sum of the interior angles because we can see that each triangle has one of its corners at the center.

We'd clearly be over-counting by including those center angles. Let's notice something though. Because all of these triangles meet at the center, we can draw a full circle out of the angles where they meet. It is a bit difficult but I think you are smart enough to master it. Let x n be the sum of interior angles of a n-sided polygon.

So you may say that x n-1 is the sum of interior angles of an n-1 -sided polygon. As in the diagram, if you cut away one vertex, say A 1 , of an n-sided polygon you can get an n-1 sided polygon, A 2 A 3 A 4 …A n. So, you get the difference equation:. Lastly, we get the angle sum of triangle. Adding up all the n-2 equalities, and canceling all the terms, we get. Insight Wow! The sum of all the angles in all the triangles equals the sum of the interior angles of the polygon. One way to understand this is to look at the exterior angles in a traverse one way around the polygon - the exterior angle being the angle turned in passing each vertex.

If I get chance I'll add an animation or if someone wants to edit feel free. This can be extended to make the polygon grow. Hint Pick a point in the interior of the polygon.

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Learn more. This is an important fact to remember. To find the sum of the interior angles of a quadrilateral, we can use the formula again. This time, substitute 4 for n. We find that the sum of the interior angles of a quadrilateral is degrees. Polygons can be separated into triangles by drawing all the diagonals that can be drawn from one single vertex.

Let's try it with the quadrilateral shown here. From vertex A, we can draw only one diagonal, to vertex D. A quadrilateral can therefore be separated into two triangles.