Interior angles

Question

Determine the number of sides in a polygon if the sum of the interior angles is 5400^o

Answer 1

The underlying principle you need to solve this question is how many of the simplest polygons can you subdivide this polygon in question into. 

 

The simplest polygon that exists is a triangle because you need at least 3 sides to enclose a finite non-zero area. And in a triangle, the interior angles sum to 180. 

 

So imagine you take a polygon pick any vertex and draw lines to connect it to every other vertex that's not immediately next to it. You'll have subdivided the polygon into triangles, each of which has an interior angle sum of 180. 

 

If you look at what you drew, you'll also realize that every angle in that original polygon is also subdivided by angles of the triangles you drew. So if you multiplied 180 by the number of triangles contained in the polygon, that will give you the sum of all the interior angles in the polygon. 

 

So for this problem, you first need to figure out how many triangles can be contained in the polygon:

 

5400/180 = 30

 

OK, so you have 30 triangles, how do you correlate that to the number of sides. You look at your drawing and realize that except for the 2 triangles which contain the 2 sides immediately next to the vertex that you originally started with every other triangle shares only 1 side with the original polygon. The 2 triangles containing the sides next to the original vertex shares 2 sides. 

 

Thus, if you have N triangles in a polygon, the number of sides will be:

 

(N-2) + 4 = N+2

 

Thus for a polygon containing 30 triangles, it will have 32 sides.