Counting Diagonals
The Goal:
As the number of sides increses in polygons, drawing and counting diagonals becomes may be very difficult.
In this activity, you will discover a logic to calculate the number of diagonals using Combinations, without counting them one by one.
Definitions:
- Vertices (n): The corners of the polygon.
- Edges (n): The outer lines that form the shape (black)
- Diagonals: The inner lines connecting non-adjacent vertices. (red)
- Use the Slider
to change the number of vertices.
- Observe the lines connecting every corner to every other corner.
Step 1: The "Total Connections" Idea
Set the slider to any n you want. For example: n = 5 (Pentagon).
- If we connect every point to every other point, how many lines do we get?
- We are choosing 2 points out of 5 to draw a line. That is the definition of Combination!
- The calculation above counts every line going out of a point.
- But we only want the Diagonals (the lines inside).
- Then we must remove the outer lines (the Edges).