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

How Many Diagonals are there in a 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 n^{2 }– 3n-54 = 0 n^{2 }– 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.