The diagonal of a square is a line segment that joins any two non-adjacent vertices. A square has two diagonals. Shown below is the diagonal BD of the □ ABCD.

Properties

Equal in length (congruent); in □ ABCD, AC = BD

Perpendicular to each other; AC ⊥ BD

Bisects each other; AC divides BD into OB and OD, similarly BD divides AC into OA and OC

Divide the square into 2 congruent isosceles right triangles; AC divides □ ABCD into △ABC & △ADC, again BD divides □ ABCD into △BCD & △DAB

Formula

The formula to calculate the diagonal of a square is given below:

Derivation

In □ ABCD,

Applying Pythagoras theorem in △BCD, we get

d^{2} = a^{2} + a^{2}

=> d^{2} = 2a^{2}

=> d = a × √2

Thus,

Diagonal (d) = a× √2, here a = side

Let us solve a few examples to understand the concept better.

Find the length of each diagonal of a square of side 16 units.

Solution:

As we know, Diagonal (d) = a × √2, here a = 16 units = (16 × √2) units = 16√2 units

Calculate the length of the diagonal of a square if its area is 25 square units.

Solution:

As we know, Area (A) = a^{2}, here A = 25 square units => 25 = a^{2} => a = 5 units Now, Diagonal (d) = a × √2, here a = 5 units = (5 × √2) units = 5√2 units

The length of the diagonal of a square is 4√2 units. What is the length of its side length?

Solution:

As we know, Diagonal (d) = a × √2, here d = 4√2 units => 4√2 = a × √2 => a = 4 units Thus, the length of each side of the square of diagonal 4√2 units is 4 units