Why does 180 n 2 work

Therefore, the sum of interior angles of a polygon having n sides is 2n — 4 right angles. Now we will learn how to find the find the sum of interior angles of different polygons using the formula. Number of Sides. Sum of interior angles 2n - 4 right angles. Solved examples on sum of the interior angles of a polygon:. Find the sum of the measure of interior angle of a polygon having 19 sides. S olution:. We know that the sum of the interior angles of a polygon is 2n - 4 right angles.

Substitute 3 for n. We find that the sum is degrees. 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. 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! What theorem can you see from the drawing?

Sum of Interior angles of an n-sided polygon There are many methods to find the sum of the interior angles of an n-sided convex polygon. Method 1 From any one of the vertices, say A 1 , construct diagonals to other vertices. 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 degrees, and so, if we subtract , we'll be left with just the angles at the edges.

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.

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.