Diagonal of Square Formula

First, let us see what is diagonal? before we get to know what is the diagonal of square and rectangle formula? A line segment that is obtained by joining any of the two non-adjacent vertices on a plane is known as the diagonal of a polygon. The diagonal of the square formula can be used to measure the length of the diagonal when the length of a side is given. In geometry, a rectangle is considered a regular quadrilateral and all four angles measure \[90^{\bullet}\]. If all the sides of the rectangle are equal then it is called square. Hence the sum of all the internal angles present in a square is equal to \[360^{\bullet}\].

A square consists of two diagonals. Where each of these diagonals is formed by joining the opposite vertices of a square diagonally. The properties of diagonals are as follows-

• Both the diagonals in a square are congruent which means the same length.

• These two diagonals can bisect each other, the point at which these two diagonals bisect is known as the midpoint of both the diagonals.

• A diagonal can divide a square into two isosceles right-angled triangles.

Diagonal Formula of Square

The below-given diagram represents a square, where AC and BD are the diagonals of a square. The formula for finding the diagonal of a square can be derived by using the Pythagoras theorem.

(Image)

Diagonal formula for square,  \[d = x \sqrt{2}\]

Where, ‘d’ represents the diagonal of the square, ‘x’ represents the side of the square.

Derivation of Formula to Find Diagonal of Square

 Let us see how to derive the formula for the diagonal of the square.

The length of both the diagonals of a square is the same. The length of a diagonal of a square is represented as ‘d’, the length of a side of a square is represented by 'x'. The diagonal formula for squares can be calculated by using the Pythagoras theorem. 

(IMAGE)

In the above diagram, consider the triangle ADC present in the square.

By using the Pythagoras theorem we can write,

We can write it as,  \[d= \sqrt{(x)^{2}+(x)^{2}}\] 

By simplifying this can be written as, 

\[d= \sqrt{2(x)^{2}}\]

\[d =  \sqrt{2}{x}\]

Area of Square Formula with Diagonal

The area of a square can be found by using its diagonals. Along with the traditional method that is used to find the area of the square by squaring its sides, there is another most useful method to find the area of a square by using the length of a diagonal.

\[Area~of~square = \frac{1}{2}d^{2}\] square units

Here, d represents the length of the diagonal.

(Image)

In the above given figure, ‘x’ unit is the side of a square, a square can be divided in the form of two right angled triangles by drawing a diagonal from the vertices A to C, say diagonal is of length ‘d’ units. Consider any of the right angled triangles and apply the Pythagoras theorem to it.

According to Pythagoras theorem, for any right-angled triangle we can write,

\[Hypotenuse^{2} = Perpendicular^{2} + Base^{2} \]

Since all the sides of a square are equal, Perpendicular and Base will be ‘a’, and the Hypotenuse will be ‘d’, 

\[ d^2=x^2+x^2\]

  \[ d^2=2 x^2\]

  \[ x^2= \frac{1}{2} {d^2}\]

So, the area of square using diagonal formula =  \[ \frac{1}{2} {d^2}\] Square units.

Solved Examples with Step by Step Solution 

Example 1: Find the length of each of the diagonals present in the square that has a side of length 14 units. Round of your answer to the nearest tenth place.

Solution: The diagonal of a square with side 14 units is to be found, 

The length of diagonal of square formula is given as \[d =  \sqrt{2}{x}\]

Here ‘x’ represents the side of a square with value 14 units, when we substitute it we get,

  \[d =  \sqrt{2}{14}\]

d=19.799

Since it is mentioned in the question that the final answer should be rounded off to tenth place we get the final answer as,

d = 19.8 units

Example 2: Find the length of a diagonal of the square if the area of a square is given as 36 square units.

Solution: The diagonal of a square can be found by using the diagonal of a square formula by area, where area of the square is given as 36 sq units.

We know that, \[ x^{2}=1/2d^{2}\] ------ (1)

Here x2=36 square units

\[x =  \sqrt{36}\]  units 

x=6 units

From equation (1) we can write the formula for finding diagonal of square can be written as,

  \[d =  \sqrt{2}{x}\]

\[d =  \sqrt{2}{6}\]

d=8.49 units

Multiple Choice Questions 

Q1. The area of a square having a diagonal of length 20 cm is? 

1.  50  \[cm^{2}\]

2. 100 \[cm^{2}\]

3.  200 \[cm^{2}\]

4. 300 \[cm^{2}\]

Solution : It is given that d = 20 cm.

We know that are of the square = ½ x  \[diagonal^{2}\] = ½ (20)2 = 200 \[cm^{2}\]

Hence the correct answer is option C.

Q2. The length of the diagonal of a square with side 14 units 

1.  25.45 units 

2.  28.65  units 

3.  32.45 units 

4. 20.22 units 

Solution: It is given that the side of the square is : a = 18 units .

We know that diagonal formula as , 

\[d =  a\sqrt{2}\] = 18 2 = 25.45 units 

Hence the correct answer is option (A)

Conclusion 

The diagonal of a square is said to be  line segment that joins two non adjacent vertices.  A square has two equal-length diagonals that bisect each other at right angles. When the side length of a square is known, the diagonal of square formula is used to compute the length of the diagonal.Throughout the article we have learned the diagonal of the square formula , its derivation and also few examples were solved. Students can learn more about diagonal of the square and other topics related to it on the official website of Vedantu.