Area of Square Using Diagonal

Area of any geometrical figure is the space occupied by a two-dimensional object. A square is a two-dimensional geometric shape that is determined by the sides which are all equal in length and perpendicular to each other (angle between two sides is 90 degrees). The area of a square is the number of square units required to fill a square fully. There are different ways to calculate the area of a square. 

One of the conventional and standard ways to calculate the area of the square is by using its diagonals or by using its sides. Since all the sides of a square are the same, we can directly find the square of its side. Therefore, the area of a square is equal to the product of any of its two sides. 

But sometimes the length of the side is not given and all we know is the length of the square's diagonal. With the knowledge of right triangles, we can find the area of a square using diagonal. 

What is the Diagonal of a Square?

A diagonal is a line that stretches from one corner of a figure to the opposite corner, passing through the center of the figure. The diagonals of a square are always equal to each other. In a polygon, the diagonals can be defined as a line joining its two nonadjacent vertices. 

The Relation Between Diagonal and Side of Square

A square can be divided into two right triangles, where the diagonal of the square is equal to the length of the hypotenuse of the triangle. Pythagoras theorem, which applies to right-angled triangles, shows the relation between the hypotenuse and sides of a right triangle.

Thus, it also represents the relation between the diagonal of a square (the hypotenuse of the triangle) and its sides. 

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Using Pythagoras Theorem,

(Hypotenuse)²= (Base)²+ (Perpendicular)²

Here, the length of the base is equal to the length of perpendicular which is denoted by ‘a’ and hypotenuse is equal to the diagonal which is denoted by ‘d’.

Therefore, a²+ a² = d²

Diagonal = \[\sqrt{a^{2} + a^{2}}\]

= \[\sqrt{2a^{2}}\]

= \[a\sqrt{2}\]

= side \[\sqrt{2}\]

Formula of the Area of Square Using Diagonal

Using the length of the diagonal, the area of a square can be calculated as:

Area of square = ½ × d² units²

Here, “d” is the length of any of the diagonals. Also, remember that in a square, diagonals are equal.

Derivation of the Area of Square Using Diagonal

We know the formula to find the area of a square using diagonals. Now, we will derive that formula using the following two methods. 

1. Using Pythagoras Theorem

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In the given figure, the diagonal of length “d” units divide the square of the side “a” units into two right triangles. Now, applying Pythagoras theorem in any right-angled triangle,

(Hypotenuse)²= (Base)²+ (Perpendicular)²

Here, 

Perpendicular = a

Base = a

Hypotenuse = d

So,

a²+ a² = d²

2a² = d²

Or, a² = d²/2

We know that area of a square = a² = d²/2

Thus, area of a square using diagonals = ½ × d² square units.

2. Using Relation between Side and Diagonal

For a square of side length ‘a’ and diagonal length ‘d’, we know,

Area of a square = side x side = a²

Now, as we have derived above,

Diagonal of square = side x √2 = a√2

Then, side of square, a = 1/√2 x diagonal = d/√2

Thus, area of square  = a²

Area =  (d/√2)²

Area = d²/2

Area = ½ x d²

Area = ½ x (diagonal)²

Thus, area of a square using diagonals = ½ × d² square units.

Solved Examples

Example 1: Find the sides and area of a square when diagonal is given as 6 cm.

Solution: Let us take a square of side x. If the square is divided into two right-angled triangles then the hypotenuse of each triangle is equal to the diagonal of the square. As given, the diagonal is equal to 6 cm.

According to Pythagoras theorem,

x² + x² = 6² 

2x² = 36

x² = 18

x =\[\sqrt{18}\]

x = 3\[\sqrt{2}\] units

Thus, the length of the side of a square is 3\[\sqrt{2}\]  units.

To find the area of a square, when diagonal is given, we can use any of the below methods:

1. Method 1 

 Area of a square = side x side =3\[\sqrt{2}\]  x 3\[\sqrt{2}\]

 = 9 x 2 = 18 cm²

2. Method 2

Area of a square = ½ x d² =  ½ x 6 x 6 

=  ½ x 36 = 18 cm²

Example 2: Find the length of the diagonal of a square using the Pythagoras theorem if the sides are 4 cm.

Solution: We know that all the sides of a square are equal in length. We also know that each vertex makes an angle of 90°. Now, let’s split the square into two right triangles, with sides equal to 4 cm. Using the Pythagoras theorem in one of the triangles, we will find the third side of the triangle, which is the diagonal of the square.  

Let the hypotenuse/ diagonal be ‘c’ cm. 

Therefore, (Hypotenuse)²= (Base)²+ (Perpendicular)²

 c²= 4² + 4² 

 = c²= 16 + 16 

c = \[\sqrt{32}\] cm

c = 4\[\sqrt{2}\]cm

The length of the diagonal is 4\[\sqrt{2}\]cm