Worksheet: Sum of the angles of a polygon. . . . . . . . . . . . . name(s) ________________________

What is the sum of the angles of a polygon? Let's explore this question, using inductive reasoning. Inductive reasoning is a type of reasoning in which, rather than proving a theorem, we solve a series of related numerical problems, notice patterns, and make a conclusion based on the patterns that we find.

For example, what number comes next in this sequence: 2,4,8,16,32 . . . ?

(Answer the question before you read on.)

ANSWER: if you answered "64", then you are correct. The pattern is "powers of 2" or "multiply the previous number by 2 to get the next number". And you probably figured it out by looking for a pattern, trying some things, then checking to see if what you tried worked. This is called "inductive reasoning".

In this worksheet, we will use deductive reasoning to find out what the sum of the angles of a regular polygon might be, and derive a formula that will work for a polygon with any number. In our formula, "n" will represent the number of sides.

With a pencil, triangulate each of the polygons drawn in the chart below. Triangulate means to connect vertices in such a way that the polygon is made up of triangles, as few triangles as possible. It is easiest to start at one vertex, and continue from there. For example, if we were to triangulate an eight sided polygon (octagon), it would look like this:

Now, you probably already know that the sum of the angles of a triangle is 180°, so you can go on to fill in the chart below. Look for patterns in your answers, and try to figure out, using inductive reasoning, what the sum of the interior angles of an "n-gon" (a polygon with "n" sides) would be. By the time you finish, you should know a lot about the angles of polygons!