Number of Diagonals

A diagonal of a polygon is a line segment that connects two vertices that are not next to each other.

Use the canvas below to find the total number of line segments, and the number of diagonals for each polygon.

How many segments will there be in a 20-sided polygon (icosagon)? How many of them will be the diagonals of the icosagon? Try to write a general term for the total number of line segments and the number of diagonals of any polygon.

Thinking about these questions can help you to find a general rule about the number of diagonals.

• What is your strategy to count the total number of segments for each polygon? What did you realize?
• Is there a number pattern for the total number of segments? How many of them are the sides and how many are the diagonals?
• What is the number of segments you draw from each vertex? How is it related to the total number?
• What is the number of diagonals you can draw per vertex?
• How can we avoid double counting when finding the total number of diagonals?

Solution

There are several ways of writing a general rule for an n-sided polygon.

You may recognize the pattern for the total number of segments:

3-6-10-15-21-28-36-45 ..

It is the sequence of triangular numbers starting from 3.

One way to express the total number of segments for an n-gon is $n(n-1)/2$ . Since there are n sides, the remaining $n(n-1)/2 -n$ of them are the diagonals of a convex polygon.

Another way to express the general rule for the total number of diagonals is to think about the number of diagonals that can be drawn from each vertex in a polygon.

That is "3" less than the number of sides.

To find the number of diagonals in a polygon, we multiply the number of diagonals per vertex $(n-3)$ by the number of vertices, $n$ , and divide by 2 (otherwise each diagonal is counted twice);

$n(n-3)/2$

Therefore, for a 20-sided polygon, there will be 190 lines and 170 diagonals.