# Diagonals of a Polygon

The diagonal of a polygon is the line segment that links the opposite, nonadjacent corners or vertices of that polygon.

## How to Find the Number of Diagonals in a Regular Polygon

For an n-sided regular polygon, the number of diagonals can be obtained using the formula given below:

This equation is obtained by adding the number of diagonals that each vertex sends to another vertex and then subtracting the total number of sides from it.

For example, in a pentagon the total number of sides is five. Hence, the number of diagonals in them are 5(5-3)/2 = 5

Let us understand this concept using examples.

Find the number of diagonals in a 15-sided regular polygon.

Solution:

As we know,
Number of diagonals = n (n – 3)/2, here n = 15
= 15(15-3)/2
= 90

In a 12-sided polygon, one vertex does not have any diagonals. Calculate how many diagonals the 12-sided polygon contains.

Solution:

As we know,
In a 12-sided polygon, the total number of diagonals = n (n – 3)/2, here n = 12
= 12(12-3)/2
= 54
Now, since one vertex does not have any diagonals, the number of diagonals of that vertex needs to be subtracted from the total number of diagonals.
In a polygon each vertex makes (n-3) diagonals, in this 12-sided polygon each vertex makes (12-3) = 9 diagonals
Hence, the total number of diagonals in this polygon is = (54-9) = 45

A polygon has 27 diagonals. Find the number of sides present in the polygon.

Solution:

As we know,
Number of diagonals = n (n – 3)/2
Now, according to the question,
n (n – 3)/2 = 27
n(n – 3) = 54
n2 – 3n-54 = 0
n2 – 9n + 6n-54 = 0
n(n-9) – 6(n-9) = 0
(n+6) (n-9) = 0
n = 9, -6
Since, n is a negative value, the number of sides present in the polygon is 9.

## FAQs

Q1. Are lines of symmetry also diagonals in a polygon?

Ans. No, not all diagonals are lines of symmetry in a polygon. For example, a rectangle has two diagonals that are not lines of symmetry, whereas in a square the diagonals are also its lines of symmetry.

Q2. What polygon has perpendicular diagonals?

Ans. A square, rhombs, and kite are polygons that have perpendicular diagonals.