Diagonals

What is Diagonal?

A diagonal is defined as a line segment joining the two opposite vertices of a polygon. Here you can read about diagonals, the formula to calculate the number of diagonals, diagonal of square formula, diagonal of rectangle formula, diagonals of rhombus and parallelogram, and some fun properties of diagonals.

Basic Terms:

• Polygon: 

It is a plane figure with three or more sides and angles. The corners of a polygon are connected by line segments. Examples- square, triangle, rectangle, decagon. A circle does not classify as a polygon.

• Vertex: 

(plural- vertices) It is described as the corner of any shape.

The number of diagonals of a polygon depends on the number of sides it has. There is a simple formula to determine the number of diagonals in a polygon.

Number of diagonals= (n(n-3))/2; where n is the number of vertices of the polygon

Example- To calculate the number of diagonals of a hexagon, we take n=6 (because it has 6 vertices)

Number of diagonals= (6(6-3))/2 = (6*3)/2 = 18/2= 9.

Therefore, there are 9 diagonals present in a hexagon. Similarly, we can calculate for the other polygons.

Why Do We Need Diagonals?

Diagonals make the framework or base for any structure. We put rods diagonally in an empty frame to stabilize it and keep it from breaking. Huge steel rods are also placed diagonally in very tall or heavy buildings to prevent them from falling or crashing on itself. You will use diagonals not only in Math but also in your everyday life!

Diagonal of Square:

A square has two diagonals. The diagonal of a square is the line segment that is drawn between its opposite corners.

Properties of Square Diagonals:

• The diagonals of a square are of the same length. Thus, they are also called congruent diagonals. 

• Each diagonal bisects the other. The diagonal is divided into two equal parts when the diagonals intersect each other.

• Each of the diagonals divides the square into two congruent isosceles right triangles because when the diagonals intersect each other, they do so perpendicularly. Therefore, the two right triangles thus obtained have the same area and each triangle has half the area of the square. The length of the diagonal of the square is also the length of the hypotenuse of the right triangle.

The length of the diagonal of a square can be found easily if we have the length of its side. To find the diagonal of a square, we use the given formula.

The diagonal formula of square =a/sqrt (2); where a is the length of any one of the sides of the square. 

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Diagonal of Rectangle:

Rectangles are similar to squares; therefore, they also have two diagonals. Their diagonals are also congruent to each other and bisect each other. Two congruent right triangles are obtained if the diagonal bisects a rectangle.

To find the diagonal of the rectangle provided that we have the dimensions of its sides, we can use the diagonal formula of the rectangle.

Diagonal of rectangle= sqrt (a2 + b2), where ‘a’ is the length of the rectangle and ‘b’ is the breadth (width) of the rectangle.

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Diagonal of Rhombus: 

A rhombus has four sides with two diagonals. When all the angles of a rhombus become 90 degrees, then the rhombus turns into a square. 

Area of rhombus= A= (PQ)/2; where ‘p’ and ‘q’ are the lengths of diagonals of the rhombus

 The formula for the length of one diagonal of a rhombus is: p= 2(a)/q

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Diagonal of Parallelogram: 

A parallelogram is a rectangle with its angles not equal to 90 degrees. The two diagonals of a parallelogram bisect each other, and the opposite sides and angles of any parallelogram are congruent.

The length of one diagonal of a parallelogram can be determined if the dimensions of its sides and the length of the other diagonal is provided.

Length of diagonal = p = sqrt (2a2 + 2b2 – (q)2)

where ‘p’ is the unknown diagonal length, q is the given diagonal length, ‘a’ and ‘b’ are the length and breadth of the parallelogram, respectively.

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Fun Fact: 

A diagonal can also be situated outside the polygon. This happens in the case of some concave polygons.

A Few Example Problems:

1. Find the Length of a Square Diagonal if One of its Sides is 8 cm.

The length of the side, a= 8 cm 

Length of diagonal = a/sqrt (2) = 8/sqrt (2) = 5.66 cm

2. Given that the Dimensions of a Rectangle are 3cm x 4cm, Find the Length of its Diagonals.

Given length l= 3 cm; breadth b= 4cm

Length of diagonal= sqrt (l2 + b2) = sqrt (32 + 42) = sqrt (9 + 16)

= sqrt (25) = 5 cm

So, the length of the diagonal is 5 cm.