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
Diagonal of 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:
Formula for Number of Diagonals of a Polygon
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.
