Diagonal of Square

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

1. Equal in length (congruent); in □ ABCD, AC = BD
2. Perpendicular to each other; AC ⊥ BD
3. Bisects each other; AC divides BD into OB and OD, similarly BD divides AC into OA and OC
4. 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

d² = a² + a²

=> d² = 2a²

=> 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) = a², here A = 25 square units
=> 25 = a²
=> 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