 # Diagonals of a Polygon: Explained in Detail

Mathematics is an interesting subject, where you will learn to calculate difficult sums and learn about new shapes and sizes. In geometry, a polygon is a simple figure enclosed by straight lines. According to Greek, poly means many, and gon means angle; like the Greek name, a polygon has many angles.

An example of the simplest polygon will be a triangle having three sides and three angles. There can be enumerable-sided polygons. If the sides and interior angles are equal, then it is a regular polygon. On the other hand, a polygon with unequal sides and angles is called an irregular polygon.

Based on the interior angle, it can be further classified into concave and convex polygons. If the interior angle is more than 180 degrees, then the polygon is known as a concave polygon and, if the interior angle is less than 180 degrees, then it is known as a convex polygon. Let us learn more about the diagonals of the polygon.

## How many diagonals are present?

Before we learn about the diagonals of polygons, we need to know how to determine the Number of Diagonals in a Polygon. A diagonal is a line segment that helps to join two vertices. You will never find diagonals on either side of a vertex as that will be on the side; hence.

Therefore, to determine the number of diagonals present in a structure, it will always be three less than the number of sides, i.e., if n is the number of vertices then, diagonal = n – 3.

If there are n number of vertices present, then the number of diagonals present is n(n – 3).

If you want to find out the number of diagonals present in a polygon, we know that every polygon has two ends, so the number of diagonals counted will be twice. Hence, we need to divide the formula by 2.

The total number of diagonals present in a polygon = n(n – 3)/2

## About the Diagonals of Polygon

Polygons can have an infinite number of sides; hence the diagonals of the polygon are ends of non-adjacent sides of a polygon. Let us determine the number of diagonals present in n-polygon.

• Triangle

Let us determine the simplest polygon, triangles which have three vertices, i.e. n = 3. As you can see, the triangle has no diagonals as there are only adjacent vertices. The number of diagonals in a triangle is 0. If you want to use the formula and want to determine: n( n – 3 )/2 = 3( 3 – 3 )/2 = 0

Now, for a quadrilateral polygon, which has 4 vertices, i.e. n = 4, the number of diagonals = 4( 4 – 3 )/2 = 2.

Therefore, the number of diagonals present in a quadrilateral is 2.

• Pentagon

For a pentagon that has 5 sides, n =5, the number of diagonals present = 5( 5 – 3)/2 = 5.

The total number of polygons present in a pentagon is 5.

• Hexagon

For a hexagon that has 6 sides, n= 6. The total number of diagonals present = 6( 6 – 3)/2 = 9.

Therefore, the total number of diagonals present in a hexagon is 9.

## Formulas

To solve any problem with diagonals, we need to learn some new formulas. Here are some of the important formulae:

• The formula of diagonal of Square

As a square has the same vertices of the same length. It is very easy to determine the diagonals of a square. Let us take a side of the square.

Diagonal of square = a√2

• The formula of diagonal of Rectangle

Two sides or vertices of a rectangle are the same. Let us take l as the length of a rectangle and b as the breadth of the rectangle.

Diagonal of a rectangle = √(l2 + b2)

• The formula of diagonal of Cube

As a cube is a three-dimensional structure, we need to consider the three-axis present. To get the diagonals of a cube, we need to use the formula of  Pythagoras Theorem. Let s be the vertices of the cube,

Diagonal of the cube = √(s2 + s2 + s2)

## Solved Examples

Let us solve some simple problems so that our concept about the diagonals becomes clear.

1. In a square, the length of the vertices is 12 cm, respectively. Find out the length of the diagonal?

Ans: Given the length of the vertices of the square, a = 12 cm.

Let d be the length of the diagonal of the square.

Using the formula of diagonals of the square, d = a√2 = 12√2 cm

The length of the diagonal of the square is 12√2 cm

1. Find out the total number of diagonals present in a 12-sided polygon?

Ans: The number of vertices present in a 12 – sided polygon, n = 12

Using the formula,

Total number of diagonals = n(n – 3)/2 = 12(12 – 3)/2 = (12 x 9)/2 = 108/2 = 54

The total number of diagonals present in a 12 – sided polygon is 54.

1. Find the length of a diagonal of a rectangle with a length of 6 cm and breadth of 4 cm?

Ans: Given the length of the rectangle, l = 6 cm

The breadth of the rectangle, b = 2 cm

Let d be the length of the diagonal of the rectangle.

Using the formula of diagonals of the rectangle, d = √(l2 +b2) = √(62 + 22) = √16 = 4 cm

The length of the diagonal of the rectangle is 4 cm.

## Conclusion

Here we learned about the various ways to find out the number of diagonals present in a polygon. We also learn about how to determine the length of the diagonal present in a polygon. We need to remember the formulas and apply them while solving any problem. Hence, we should practice solving more problems to avoid facing any difficulty during examinations.

## FAQs

1. What is a 10 sided polygon called?

Ans: A 10-sided polygon is called a Decagon.