Question

The diagonals of a polygon are lines that join any two nonadjacent vertices. The square has two diagonals, the pentagon has five, and the hexagon has nine. The equation for the number of diagonals of any n-sided polygon is

Answer 1

Here's how the equation for the number of diagonals is derived:

Let's say we have a 9-gon.

Square the number. This way each vertex (1, 2, 3...9) has one diagonal going to each other vertex counted.

Instead of using n × n, use n × (n - 3).

This way a vertex can not make a diagonal with itself or either vertex adjacent (simply remove three of the possible diagonals, right?)

Divide that whole thing by 2, since a diagonal from vertex 2 to 7 is the same as one from 7 to 2. (This eliminates reverses of the same diagonal)

where d = diagonals and n = sides

Explanation:

Answer 2

Here's how the equation for the number of diagonals is derived:

Let's say we have a 9-gon.

Square the number. This way each vertex (1, 2, 3...9) has one diagonal going to each other vertex counted.

Instead of using n × n, use n × (n - 3).
This way a vertex can not make a diagonal with itself or either vertex adjacent (simply remove three of the possible diagonals, right?)

Divide that whole thing by 2, since a diagonal from vertex 2 to 7 is the same as one from 7 to 2. (This eliminates reverses of the same diagonal)


`\boxed{d=\frac{n(n-3)}2}` where d = diagonals and n = sides