Diagonals of Polygon Formulas

In Geometry, diagonals of a polygon are line segments joining two non-adjacent vertices.

Example:

A square given below has two pairs of non-adjacent vertices.

By joining the vertices of these two pairs, we get two diagonals of a square.

In this square ABCD, A and C, and B and D are two pairs of adjacent vertices.

Hence, AC and BD are diagonals of a square.

The formula to find the diagonal of a polygon square is:

\[\frac{n(n-3)}{2}\], where n is the number of sides.

We can use this formula to find the diagonals of a polygon with any number of sides.

In this article, we will discuss the diagonals of a polygon formula and the formula to find the number of diagonals in a polygon.

What is a Polygon?

A simple closed figure made up of line segments is known as a polygon. Polygons are generally classified according to the number of vertices and sides they have. For example, a triangle is a polygon with 3 sides and 3 vertices, similarly, a quadrilateral is a polygon with 4 vertices and 4 sides.

Formulas for Number of Diagonals in a Polygon

Can you think of the number of diagonals of a quadrilateral, heptagon, hexagon, octagon, nonagon, etc?

You can find the number of diagonals of different polygons using the number of diagonals formula.

As you can see, it is quite difficult to draw diagonals of a polygon with a large number of sides.

To easily find the number of diagonals of a polygon with a large number of sides, we can use the formulas to find the number of diagonals in a polygon with n sides without actually drawing them.

Number of Diagonals Formula

The diagonal of a polygon is a line segment obtained by connecting two opposite angles or non-adjacent vertices. The number of diagonals and their properties are different, based on the number of edges, based on the type of polygon. If the number of sides of a polygon is n, the number of diagonals that can be displayed is given by n (n – 3) 2.

This formula can be used to calculate the number of diagonals of a polygon. It depends on the number of sides and the type of polygon. You can use this formula to calculate the number of diagonals  without drawing polygons. 

The formula to find diagonals of a polygon with n side is:

The following table shows the number of diagonals of different polygons which is calculated using the formula of diagonals of a polygon.

Number of Diagonals of Polygons Table

Solved Examples Using Formula for Number of Diagonals in a Polygon

1. Find the number of diagonals of a hexadecagon.

Solution:

The number of sides of an hexadecagon, is n = 16

The number of diagonals of octagon can be found by using the no. of diagonals formula

\[\frac{n(n-3)}{2}\]

As we know, the total number of sides of a hexadecagon is 16. Substituting the values in the formula of diagonal of polygon we get,

\[\frac{n(n-3)}{2}\]

= \[\frac{16(16-3)}{2}\]

= \[\frac{16(13)}{2}\]

= \[\frac{(208)}{2}\]

= 104

The number of diagonals of an octagon = 104.

2. Find the number of sides of a polygon with 90 diagonals?

Solution:

Let the number of sides of a polygon with 90 diagonals = x

Total number of diagonals = 90

The number of sides of of a polygon with 90 diagonals can be found by using the number of diagonals formula

\[\frac{n(n-3)}{2}\]

As we know, the total number of sides of a hexadecagon is x. Substituting the values in the formula of diagonal of polygon we get,

\[\frac{n(n-3)}{2}\]

\[\frac{x(x-3)}{2}\] = 90

\[\frac{(x^{2}-3x)}{2}\] = 90

x\[^{2}\]-3x = 180

x\[^{2}\]-3x-180 = 0

= (x-15) (x+12)

x=15, x=-12

As the value of x cannot be negative. Hence, the number of sides of a polygon with 90 diagonals is 15.

The number of sides of a polygon with 90 diagonals = 15.

Polygon

A closed curve or entity formed by a line segment in such a way that two lines do not intersect, except that their endpoints meet to form a closed figure is called a polygon. In other words, a polygon is a simple closed curve consisting only of  line segments where each line is its sides. Polygons with all the same sides and angles are regular polygons.In Greek, Poly is defined as many and Gon is defined as angle.

Shapes such as triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, and decagon are called a polygon accordingly as it comprises of 3,4,5,6,7,8,9,10 number of sides respectively. 

Types of Polygon

Polygons are of 4 kinds:

1. Convex Polygon 

Polygons with each angle less than 180 degrees are called convex polygons.

2. Concave

Polygons with at least one angle more than 180 degrees are called concave polygon

3. Regular

Polygons, where all sides and angles are equal, is known as Regular Polygon

4. Irregular 

Polygons where sides and angles are not equal or when the polygon is not regular then it is known as an irregular polygon.

Diagonal

The line that connects the vertices to the non-adjacent vertices is called the diagonal of the polygon. The simplest polygon which has no diagonals is Triangle. You cannot draw a line from one internal angle of a triangle to another. The square has two diagonals, making it the simplest form of a polygon with diagonals, Pentagons have five diagonals, whether regular or irregular, and hexagons have nine diagonals. The diagonal is always inside a  simple polygon that is convex. Consider the shape of a rectangular door. On the other side, you can draw a line from the upper hinge corner to the lower hinge corner. You can also draw a line from the bottom corner of the hinge to the top of the opposite corner. Two such diagonals are formed. The diagonals of a simple polygon with a concave surface can extend beyond the polygon, cross the sides, and be partially outside the shape. However, they are still diagonal. Concave polygons with diagonals that go beyond the shape are common in darts and stars.

The formula for Regular Polygon 

For n-sided regular polygon, the following applies: 

All outside angles = 360 °/n 

Each internal angle = 180 ° -  each external angle 

 = 180 ° -360 °/ n 

 = n × 180 ° – (2 × 180 °)/ n 

 = (n-2) x 180 °/ n 

Hence ,we can say that internal angle = (n-2) x 180 °/ n

Solved Example

1. Find the number of diagonals of a 13 sided polygon.

Ans:  n= 13

According to the given formula,

Number of diagonals= n(n-3)/2

= 13(13-3)/2 

= 65 is the total number of diagonals of a 13 sided polygon

Conclusion

We are provided with the formula to find the number of diagonals of polygons of n sides. The number of diagonals formula, given on this page helps to find the number of diagonals of a polygon of a large number of sides without actually constructing them. So, if you want to find the number of diagonals of any problem, use the formula and get the answer at no time.