# 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√2where,

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

d

^{2}= a^{2 }+ a^{2 }=> d = √(a

^{2}+ a^{2})=> d = √(2a

^{2})=> d = √2 × √a

^{2}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