why do you have to subtract 2 in the formula (n-2)180 (this is the sum of interior angles of polygons formula).

Question

 I am trying to explain the formula (n-2)180 used to find the sum of interior angles of polygons, I understand the formula but I'm just wondering why it's minus 2? I know that you can also split the polygon into 3 triangles and add up the sums to get 540 but why 3 triangles? Why not 5 triangles?

Answer 1

It's true that it's confusing. If you're drawing the triangles all coming from the center, the thing to realize is that the (n-2) isn't because there are n-2 triangles. There are clearly n triangles when you draw it out that way. Instead, the -2 comes out of the math.

 

Let's consider what happens when you draw these triangles. They all come out from one point in the center, n in total. The total value of the angles in all these triangles is 180*n. 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. That means that all of these angles in the center add up to 360 degrees, and so, if we subtract 360, we'll be left with just the angles at the edges. That means that our sum will be n*180 - 360 = n*180 - 2*180 = (n-2)*180

 

 

It's worth noting, there's another way to draw the triangles that does create just n-2 triangles, with all their angles at the polygon's vertices. Take a side of your polygon, and draw a line from one of that side's vertices to the other's neighbor. (For instance, if you have a pentagram ABCDE, take side AB and add line AC.) Now, from that new vertex draw a line to the neighbor of the vertex you just connected it to. (Connect C to E) Repeat until you'd be connecting a vertex to its own neighbor (which it's already connected to, of course). This method does lead to an intuitive (n-2)*180