In a polygon, a diagonal is a segment that connects two non-consecutive vertices. In this part of the resource, you will develop an algebraic expression that can be used to determine the number of diagonals in a polygon if you know the number of sides the polygon has.

The dynamic geometry sketch opens with a regular pentagon, and counts the number of diagonals for you in the bottom right portion of the sketch. Use this sketch to fill in a table like the one shown below. You may create the table in your notes.

Number of Sides Number of Diagonals
3
4
5
6
8
10
s d

Before recording the number of diagonals for a polygon with the given number of sides, be sure to click "Make irregular" and investigate if the number of diagonals depends on whether or not the polygon is regular. To change the number of sides in the polygon, click "more" to increase the number of sides, or "less" to reduce the number of sides.

1. What patterns do you notice in the table?
2. What algebraic expression can you write to determine the number of diagonals of a polygon with n sides?
Interactive popup. Assistance may be required.

3. Does it make a difference if the polygon is regular or irregular? Why do you think that is the case?
Interactive popup. Assistance may be required.

It does not make a difference whether the polygon is regular or irregular. The number of diagonals is determined by the number of non-consecutive vertices, and that number does not depend on side length.

4. Does it make a difference if the polygon is convex or concave? Why do you think that is the case? You may return to the sketch for additional investigation if you'd like.
Interactive popup. Assistance may be required.

It does not make a difference whether the polygon is convex or concave. The number of diagonals is determined by the number of non-consecutive vertices, and that number does not depend on side length.

### Take Notes Entry

In the formula to find the total number of distinct diagonals, n is multiplied by n – 3, then that product is divided by 2. Why do you think this is so? Record your findings in your notes.