Diagonal of a Square Formula

Square is considered to be a regular quadrilateral, with all four sides having an equal length and all equal four angles. The angles subtended at the adjacent sides of a square are right angles. In addition to this, the diagonals of the square are equal and bisect each other at 90°. 

A square is a special case of a parallelogram with two adjacent equal sides and one right vertex angle. Also, a square can be considered as a special case of a rectangle, with equal length and breadth. 

Diagonal of a Square

The diagonal is a line segment that joins any two non-adjacent vertices of the square. Both the diagonals of a square are equal to each other. It divides the square into two congruent triangles. 

Properties:

• The diagonals of a square are equal in length.
• The diagonals of a square are perpendicular bisectors of each other.
• The diagonals of a square divide the square into two congruent isosceles right-angled triangles.

Formula for Diagonal of a Square

a√2

where, 

a is the side of the square

Derivation of the Formula

Let us consider the triangle ABC in the square. We know that all the angles in a square are 90°, so by using the Pythagoras theorem, we can find the hypotenuse

d² = a² + a² 

=> d = √(a² + a²)

=> d = √(2a²)

=> d = √2 × √a²

Therefore √2a is the formula for diagonal of the square.

Sample Problems

Problem 1: Find the length of each diagonal of a square with side 6 units.

Solution:

Formula for Diagonal = √2a

√2 × 6

We know that √2 = 1.414 

Therefore, 1.414 × 6 = 8.48 units

Problem 2: The length of the diagonal of a square is 8√2 units. Find the side length of the square.

Solution:

Given the diagonal length = 8√2 so d = 8√2

therefore  

8√2 = a√2           {Using Diagonal Formula}

So side is 8 units

Problem 3: If the side of a square is 50 units, what is the length of each diagonal? 

Solution:

Given side = 50

So,

d = 50√2 

Therefore, the length of each diagonal is 50√2 

Problem 4: The length of the carrom board which is in the shape of a square is 2√2 units. Find the side length of the square.

Solution:

Given the diagonal = 2√2 so d = 2√2

therefore  

2√2 = a√2

So length of the carrom board is 2 units

Problem 5: If the side of a square field is 5 units, what is the length of each cross row?

Solution:

Here cross row means diagonal 

Given side = 5

So,

d = 5√2

Therefore, the length of each cross row is 5√2