Diagonals

A diagonal is a line segment that joins one corner to another but is not an edge. So we get a diagonal when we directly join any two corners (vertices) which are not already joined by an edge. In the case of a polygon, it is a straight line connecting the opposite corners of a polygon through its vertex.  

Table of Contents

What is a Diagonal?

A line segment joining one corner to another but is not an edge is called a diagonal. So we get a diagonal by directly joining any two corners (vertices) which are not already joined by an edge.

Shape of a Diagonal

Since a diagonal is a line segment joining non-adjacent vertices or corners, the shape of a diagonal is that of a straight line.

Diagonals of Polygons

"A diagonal of a polygon is a line segment that is obtained by joining any two non-adjacent vertices." We know that a polygon is a closed shape formed by joining the adjacent vertices. For example, a square has 4 sides, a pentagon has 5 sides, and a hexagon has 6 sides, and so on. Depending upon the type of polygon based on the number of edges, the number of diagonals and their properties would vary. Similarly, the properties of diagonals vary according to the type of solid.

Let us now understand the diagonals for different polygons. 

  • Diagonal of a Triangle
  • Diagonal of a Square
  • Diagonal of a Rectangle
  • Diagonal of a Rhombus
  • Diagonal of a Pentagon
  • Diagonal of a Hexagon

Diagonal of a Triangle

A triangle is a closed figure or shape with 3 sides, 3 angles, and 3 vertices. A triangle is the simplest type of polygon. No vertices in a triangle are non-adjacent. It means that there are no line segments that can form diagonals. 

The number of diagonals of a triangle = 0.

Diagonal of a Square

A square is a closed two-dimensional figure with four sides and four corners. The length of all four sides is equal and parallel to each other. The diagonal of a square is a line segment that joins any two of its opposite vertices. In the following square, there are two pairs of non-adjacent vertices. By joining the vertices of each such pair, we get two diagonals, AC and BD of the square. The lengths of the lines AC and BD are the same. A diagonal cuts a square into two equal right triangles and each diagonal forms the hypotenuse of the right triangles so formed. The number of diagonals of a square = 2.

Diagonals of a Square

Diagonal of a Rectangle

The diagonal of a rectangle is a line segment that joins any two of its non-adjacent vertices. In the following rectangle, AC and BD are the diagonals. You can see that the lengths of both AC and BD are the same. A diagonal cuts a rectangle into 2 right triangles, in which the sides equal to the sides of the rectangle and with a hypotenuse. That hypotenuse is the diagonal.

Diagonal of a rectangle

Diagonal of a Rhombus

The diagonal of a rhombus is that it always bisects at a 90° angle, which means that the two halves of any diagonal will be of the same length. A rhombus can be defined as a diamond-shaped quadrilateral that has all four sides equal. Unless the rhombus is a square, then the diagonals will have different values.

Diagonals of Rhombus

Diagonal of a Pentagon

A pentagon is a closed two-dimensional figure with five sides and five corners. The length of all five sides is equal in a regular pentagon. A pentagon has five diagonals as shown in the image below:

Introduction to Diagonals: A pentagon

Diagonal of a Hexagon

A hexagon is a closed two-dimensional figure with six sides and six corners. The length of all six sides is equal in a regular hexagon. A hexagon has nine diagonals as shown in the image below:

 

A hexagon has 9 diagonals

Here, all the 9 red-colored lines inside the hexagon are the diagonals.

Apart from the polygons, for different solids also, based on the number of edges, the number of diagonals and their properties would vary.

  • Diagonal of a Cube
  • Diagonal of a Cuboid

Diagonal of a Cube

A cube is a three-dimensional solid figure, also known as the square solid that has edges of all the same length. That means that the length, width, and height are equal, and each of its faces is a square. The main diagonal of a cube is the one that cuts through the center of the cube; the diagonal of a face of a cube is not the main diagonal.

Diagonals of a cube

Diagonal of a Cuboid

A cuboid is a three-dimensional analog of a rectangle in two dimensions. The main diagonal of a cuboid is the one that cuts through the center of the cuboid; the diagonal of a face of a cuboid is not the main diagonal.

Diagonal of a cuboid

Number of Diagonals Formula

The number of diagonals formula can be used to calculate the number of diagonals in a polygon. It differs according to the type of polygon, based on the number of sides. We can use this formula to find the number of diagonals of any polygon without actually drawing them:

The number of diagonals of a polygon with "n" number of sides = n(n-3)/2

The following table shows the number of diagonals of some polygons which is calculated using this formula.

Shape Number of sides, n Number of Diagonals
Triangle 3

3(3−3)/2 = 0

Quadrilateral 4

4(4−3)/2 = 2

Pentagon 5

5(5−3)/2 = 5

Hexagon 6

6(6−3)/2 = 9

Heptagon 7

7(7−3)/2 = 14

Octagon 8

8(8−3)/2 = 20

Nonagon 9

9(9−3)/2 = 27

Decagon 10

10(10−3)/2 = 35

Hendecagon  11

11(11−3)/2 = 44

Dodecagon  12

12(12−3)/2 = 54

 

Example: Find the number of diagonals of a decagon.

Solution:

The number of sides of a decagon is n = 10. The number of diagonals of a decagon is calculated using:

n(n-3)/2 = 10(10-3)/2 = 10(7)/2 = 70/2 = 35 

The number of diagonals of a decagon = 35

Length of a Diagonal

The length of a diagonal for any polygon depends upon the type of polygon. There is no general formula to calculate the length of a diagonal. Rather, based on the dimensions of the particular polygon, the formula to calculate the length of the diagonal can be found. This section will cover the formula to calculate the length of diagonal for some polygons and solids based on their structure and dimensions.

  • Length of diagonal of a square
  • Length of diagonal of a rectangle
  • Length of diagonal of a cube
  • Length of diagonal of a cuboid

Length of Diagonal of Square

In a square, the length of both the diagonals is the same. The length of a diagonal d of a square of side length x units is calculated by the Pythagoras' theorem. Using Pythagoras theorem, d = √(x+ x2) = √(2x2) = √2x units.

Length of a diagonal of a square = √2x units

Length of the diagonal of a square formula

Length of Diagonal of a Rectangle

Similar to a square, the length of both the diagonals in a rectangle are the same. The length of a diagonal d of a rectangle whose length is 'l' units and whose breadth is 'b' units is calculated by the pythagoras theorem.

Length of the diagonal of a rectangle

Using Pythagoras theorem, d2 = l+ b2

Length of a diagonal of a rectangle = √l+ b2 units

Length of a Diagonal of a Cube

Consider a cube of length x units. A cube has 6 faces. Each face of a cube is a square. Thus each face has two diagonals. Hence, the length of each such diagonal is the same as the length of a diagonal of a square. Length of each face diagonal of cube = √2x units.

Diagonals on the faces of a cube

Apart from the diagonals on the faces, there are 4 other diagonals (main diagonals or body diagonals) that pass through the center of the square. The formula for the length of the diagonal of a cube is derived in the same way as we derive the length of the diagonal of a square. Length of body diagonal of a cube = √3x units.

Length of the diagonal of a cube formula

Length of a Diagonal of a Cuboid (Rectangular Prism)

Consider a cuboid of length l, width w, and height h. Let us assume that it's main diagonal (or body diagonal) that passes through the center of the cuboid is d. Length of a diagonal of a cuboid = √(l+ w+ h2).

Length of the diagonal of a cuboid

Important Notes

  • The number of diagonals of a polygon with n sides is n(n-3)/2.
  • The length of the diagonal of a square of side x is 2.
  • The length of the diagonal of a rectangle of sides l and b is  √(l+ w2). 
  • The length of the diagonal of a cube of side x is√3x.
  • The length of the diagonal of a cuboid of sides l, w, and h is √(l+ w+ h2).

Topics Related to Diagonals

Given below is the list of topics that are closely connected to the diagonals. These topics will also give you a glimpse of how such concepts are covered in Cuemath.


FAQs on Diagonal

What is the Angle of a Diagonal in a Square?

Each diagonal in a square divides the angle at each vertex into two equal parts. Hence, the angle between any side and any diagonal is 45 degrees.

What do you Mean by Diagonals?

A diagonal of a polygon is a line segment that is obtained by joining any two non-adjacent vertices. In the case of a polygon, it is a straight line connecting the opposite corners of a polygon through its vertex. So we get a diagonal when we directly join any two corners (vertices) which are not already joined by an edge.

What does a Diagonal Look Like?

The shape of a diagonal is a line segment. It starts and ends at the two opposite vertices of a polygon.

How to Find Number of Diagonals in a Polygon?

The number of diagonals for any polygon differs according to the type of polygon, based on the number of sides. The number of diagonals formula can be used to calculate the number of diagonals in a polygon. The number of diagonals of a polygon with "n" number of sides = n(n-3)/2, where n is the total number of sides of the polygon.

How do you find the Number of Sides of a Diagonal?

A diagonal itself is a line segment. It is a line segment joining non-adjacent vertices of any shape.

How to Calculate the Length of Diagonal for any Shape?

The length of a diagonal for any polygon depends upon the type of polygon. There is no general formula to calculate the length of a diagonal. Rather, based on the dimensions of the particular polygon, the formula to calculate the length of the diagonal can be found. For example, the length of a diagonal of a square = √2x units.

How Many Diagonals Does a Triangle have? 

No vertices in a triangle are non-adjacent. It means that there are no line segments that can form diagonals. Thus, a triangle has 0 number of diagonals.


Solved Examples on Diagonal

Example 1: If a polygon has 90 diagonals, how many sides does it have?

Solution:

Let us assume that the number of sides of the given polygon is n. The number of diagonals = 90.

n(n-3)/2 = 90

n(n-3) = 180

n- 3n - 180 = 0

(n - 15)(n + 12) = 0

n = 15; & n = -12 

Since n cannot be negative, the value of n is 15. Therefore, the number of sides of the given polygon = 15.

Example 2: A cube has a wall area of 121 square units. What is the length of the main diagonal of the cube?

Solution:

Let us assume that 'x units' is the side of the given cube. Its wall is a square. Thus, the area of the wall = x2 square units. Since the area of the wall is 121 square units,

x2 = 121

x = 11

Let us assume that the length of the diagonal of the cube is d units. This implies,

d = √3x

= √3 × 11 

= 11√3 units

Therefore, the length of the main diagonal = 11√3 units.

Practice Questions on Diagonal

Here are a few activities for you to practice.

Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
  
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