Polygon Analysis

Objective

Determine a formula that calculates the number of diagonals in a convex polygon in terms of number of sides.

Base Difficulty

Easy

Concept

Any polygon with a given number of sides has a unique number of diagonals. Trying to find a relationship between the number of sides in a polygon and the number of possible diagonals involves mathematical analysis. The idea is to gather data, formulate a hypothesis based on the data, and then find a valid mathematical proof of your hypothesis.

Hypothesis:

Draw out a few polygons with their diagonals. Based on the your observations, try to come up with a few ideas. These do not have to be the exact formula itself, but what do you think the formula involves?

Materials:

Procedure:

  1. Draw out at least six polygons starting with the triangle.
  2. Draw all possible diagonals in the polygon and count the number of polygons that were drawn.
  3. Make a table with two columns. In one column, enter the number of sides in the polygon, and in the other enter the number of diagonals(Excel would be a great idea!).
  4. Based on your data, make a hypothesis.

Analysis

Did you come up with a formula? How can you prove that it works for every polygon? You might need to think in more general terms. Look at the formula you developed to see if you can find a place to start to derive the formula. You could also try induction.

Conclusion

What other meaningful results can you determine? Can you generalize the problem even more? Check the extensions below for some ideas.

Extensions

Notice that making the diagonals also partitions the polygon. Can you come up with a formula that calculates the number of partitions? A quick read on combinations might be useful.