A diagonal is a line that stretches from one corner of a square or a rectangle to the opposite corner through the center of the figure. The diagonals of a square are always equal to each other. The formula of Diagonal is also used to calculate the polygon diagonals. Diagonals can be defined as a line joining the two nonadjacent vertices of a polygon. (image will be updated soon)

As we know, the common way of finding the area of a square is by squaring the length of its sides. Thus, the measure of the length of the side of a square plays a very important role. In order to replace side by diagonal, it is necessary to find the relation between the side and diagonal of a square.

A square can be divided into two right triangles where the length of the hypotenuse of the triangle is equal to the diagonal of the square. Pythagoras theorem, which is applicable 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. (image will be updated soon)

Pythagoras Theorem

Pythagoras theorem states that the square of the hypotenuse of a right-angled triangle is equal to the sum of the square of its base and height.

(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,

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

The Relation Between Diagonal And Side of a square is,

Diagonal =\[\sqrt{a^2+a^2}\]= \[\sqrt{(2a^2)}\] = \[\sqrt{2a}\] = \[\sqrt{2}\] x side

We know,

Area of a square = side x side

If Diagonal is equal to\[\sqrt{2}\] x side

Then, side = \[\frac{1}{\sqrt{2}}\] diagonal

Thus, the formula of area of a square with diagonal = 1/√2 diagonal X 1/√2 diagonal

=1/2 (diagonal)^{2}

= 1/2 d^{2}

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

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, diagonal is equal to 6cm.

According to Pythagoras theorem,

x^{2}+x^{2}= 6^{2}

2 x^{2}= 36

x^{2}= 18

x = \[\sqrt{18}\]

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

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

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

i) Method 1 :

Area of a square = side x side = 3\[\sqrt{2}\] x 3\[\sqrt{2}\] = 9 x 2 = 18 cm.

ii) Method 2:

Area of a square = \[\frac{1}{2}\]d^{2 }= \[\frac{1}{2}\]x 6 x 6 = 12 x 36 = 18 cm

Thus, the area of a square is 18 cm, and the length of the side is 3\[\sqrt{2}\] cm.

FAQ (Frequently Asked Questions)

Question1: How to find the length of the diagonal of a square using the Pythagoras theorem if the sides are 4 centimeters?

Solution: First thing first, we are already aware that all the sides of a square are equal in lengths (i.e., 4 cm according to this case). We also know that each vertex makes an angle of 90°. Now, let the square be split into 2 Triangles so that it looks like a sandwich. Each of the straight lines is equal to 4 cm. What we need to find out is the third side of the figure of the triangle which is the diagonal. (images will be uploaded soon)

Therefore, use the formula given below:

a

^{2}+b^{2}=c^{2}4

^{2}+4^{2}=c^{2}16 + 16 = c

^{2}c = √32

c = 4√2 cm

The length of the diagonal is 4√2 cm.

Question 2: How to find the side of a square if its diagonal is given?

Solution: The diagonal in a square divides it into 2 right-angle triangles.

Since the angle of a square is 90 degrees each, we find that the two right-angle triangles obtained by fixing in the diagonal are in dimension.. 45, 45, 90 degrees. Since two angles are the same, each of them is an isosceles right-angled triangle.

Select one of the two triangles and label them as opposite and adjacent x and x (since they are equal). To calculate the hypotenuse we can use the Pythagoras theorem The value of which will be X √2. That means, if we know the hypotenuse, we only need to divide it by √2 to get any of the two sides (opposite or adjacent). We can say that the hypotenuse (diagonal of the square initially) is 10. The opposite, as well as the adjacent, will be 1/√2 each.