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 vertices.
1.  What is a Diagonal? 
2.  Diagonals of Polygons 
3.  Number of Diagonals Formula 
4.  Length of a Diagonal 
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 nonadjacent 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 nonadjacent 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 defined as a closed figure or shape that has 3 sides, 3 angles, and 3 vertices. A triangle is the simplest type of polygon. No vertices in a triangle are nonadjacent. 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 defined as a closed twodimensional figure having four sides and four corners. All the sides are parallel to each other with equal lengths. 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 nonadjacent 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 in the given square are the same. The diagonal of any square cuts it into two equal right triangles, such that diagonal makes hypotenuse of the right triangles so formed. The number of diagonals of a square = 2.
Diagonal of a Rectangle
The diagonal of a rectangle is a line segment that joins any two of its nonadjacent 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 Rhombus
The diagonals of a rhombus are the line segments joining the opposite vertices, bisecting each other 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 diamondshaped quadrilateral having all four sides equal. The diagonals of a rhombus will have different values unless the rhombus is a square.
Diagonal of a Pentagon
A pentagon is a closed twodimensional 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:
Diagonal of a Hexagon
A hexagon is a closed twodimensional 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:
Here, all the 9 redcolored 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 threedimensional 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 line segment that cuts through its center, joining the opposite vertices. While the diagonal of a face of a cube is the one joining the opposite vertices on every face. Thus, it is not the main diagonal.
Diagonal of a Cuboid
A cuboid is a threedimensional 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.
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(n3)/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(n3)/2 = 10(103)/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^{2 }+ x^{2}) = √(2x^{2}) = √2x units.
Length of a diagonal of a square = √2x units
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.
Using Pythagoras theorem, d^{2} = l^{2 }+ b^{2}
Length of a diagonal of a rectangle = √l^{2 }+ b^{2 }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.
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 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^{2 }+ w^{2 }+ h^{2}).
Important Notes
 The number of diagonals of a polygon with n sides is n(n3)/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^{2 }+ w^{2}).
 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^{2 }+ w^{2 }+ h^{2}).
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(n3)/2 = 90
n(n3) = 180
n^{2 } 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 = x^{2} square units. Since the area of the wall is 121 square units,
x^{2} = 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.
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 nonadjacent 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(n3)/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 nonadjacent 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 nonadjacent. It means that there are no line segments that can form diagonals. Thus, a triangle has 0 number of diagonals.