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Diagonal of Square
The diagonal of a square is a line segment that joins any two nonadjacent vertices. A square has two diagonals that are equal in length and bisect each other at right angles. The diagonal of square formula is used to calculate the length of the diagonal of a square when its side length is known.
What is the Diagonal of a Square?
A square has two diagonals and each diagonal is formed by joining the opposite vertices of the square. Observe the following square to relate to the properties of the diagonals given below.
 The diagonals of a square are equal in length.
 They are perpendicular bisectors of each other.
 They divide the square into two congruent isosceles rightangled triangles.
What is the Diagonal of Square Formula?
The diagonal of a square formula, is d = a√2; where 'd' is the diagonal and 'a' is the side of the square. The formula for the diagonal of a square is derived using the Pythagoras theorem. A diagonal divides a square into two isosceles rightangled triangles. Both the diagonals are congruent and they bisect each other at right angles. Let us understand how to derive the formula to find the diagonal of a square.
Derivation of Diagonal of Square Formula
In a square, the length of both the diagonals is the same. The length of a diagonal, 'd' of a square with side length 'a' is calculated using the Pythagoras theorem. Observe the following square to see that the length of the diagonal is denoted by the letter 'd' and the side length is denoted by 'a'.
Diagonal of a Square Formula
Let us consider the triangle ADC in the square. We know that all the angles in a square are 90°, therefore, using the Pythagoras theorem, we can find the hypotenuse, which is 'd' in this case.
d^{2} = a^{2 } + a^{2}
d = √(a^{2} + a^{2})
d = √(2a^{2})
d = √2 × √a^{2}
= √2a
Therefore, the diagonal of a square formula is: d = a√2
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Examples Using Diagonal of Square Formula

Example 1: Find the length of the diagonal of a square with side 14 units.
Solution:
The side length of the square is: a = 14 units.
Using the diagonal of square formula, the length of the diagonal, d is:
d = a√2
d = 14√2 or 19.8 units
Answer: The length of each diagonal of the given square is 19.8 units.

Example 2: The length of the diagonal of a square is 3√2 units. Find the side length of the square.
Solution:
It is given that the diagonal of square = 3√2 units.
Let us assume the side length of the square to be 'a'
According to the diagonal of square formula, the length of the diagonal, d is:
d = a √2
3√2 = a √2
a = 3
Answer: The side length of the given square = 3 units.

Example 3: Find the length of the diagonal of a square if its area is 36 square units.
Solution:
Area of the square = 36 square units
We know that the area of a square = a^{2} = 36
Therefore, a = 6 units
We know that the diagonal of a square formula is, d = a√2
Therefore, the length of the diagonal = a√2 = 6√2 = 8.49 units
Answer: The length of the diagonal of square = 8.49 units.
FAQs on Diagonal of Square Formula
What is the Diagonal of Square in Math?
The diagonal of a square is a line segment that joins two nonadjacent vertices. A square has two diagonals that are equal in length and bisect each other at right angles. The properties of the diagonals of a square are as follows:
 They are equal in length.
 They are perpendicular bisectors of each other.
 They divide the square into two congruent isosceles rightangled triangles.
What is the Formula of Diagonal of Square?
A square has two diagonals of the same length which can be calculated by using the formula, d = a√2, where 'a' is the side of the square.
How to Find the Diagonal of a Square Using the Diagonal Formula?
To calculate the length of the diagonal of a square we use the following steps:
 Step 1: Check for the length of the side of the square, a.
 Step 2: Place the value of 'a' in the formula for the diagonal of a square, d = a√2.
 Step 3: Write the obtained value with the appropriate unit.
How to Derive the Diagonal of a Square Formula?
The diagonal formula of a square can be derived by using the Pythagoras theorem.
 Step 1: Draw the diagonals of a square.
 Step 2: Two rightangled triangles will be formed. Consider one of the triangles.
 Step 3: Two sides of the right triangle will be the same since all the sides of a square are equal.
 Step 4: Apply the Pythagoras theorem and calculate the length of the hypotenuse of the triangle which is the diagonal of the square.
Thus, diagonal d = √(a^{2} + a^{2}) = (√2)a = a√2; where 'a' is the side of the square.
What is 'a' in the Diagonal of a Square Formula?
Since a square has four equal sides, therefore, in the diagonal of a square formula, 'a' represents the side of the square. The diagonal of a square formula is thus given as, d = a√2.
Is the Diagonal of Square Equal to its Side?
No, the diagonal of a square is not equal to its side. Since all the angles of a square are equal to 90°, the diagonal of a square becomes the hypotenuse of the triangles that are formed in the square.
How to Find the Diagonal of a Square when Area is Given?
If the area of a square is given, the side length of the square can be calculated. Then, the value of the side length can be used to find the diagonal of the square with the help of the formula, d = a√2. For example, if the area of a square is 81 square units. We will first find its side length since we know that the area of a square = a^{2}. Therefore side 'a' = √81 = 9 units. Now, we will use this value in the formula, d = a√2, d = 9√2 = 12.72 units.
How to find the Diagonal of Square when Side is Given?
The diagonal of a square can be calculated if the side is given. The diagonal of square formula = a√2; where 'a' is the side length. The given side length is substituted in this formula to get the length of the diagonal. For example, if the side length of a square is given as 10 cm, we will substitute the value in the formula, d = a√2. This means, the length of the diagonal (d) = a√2 = 10√2 = 14.14 cm.
How to find Diagonal of Square with Perimeter?
The diagonal of a square can be calculated if the perimeter of the square is given. Let us understand this using an example. For example, if the perimeter of a square is 32 units, let us find the diagonal using the following steps:
 Step 1: We know that the formula to find the perimeter of a square = 4 × side length. After substituting the given value of the perimeter, the side length of the square can be calculated. Here, this will be, Perimeter of square = 4 × side length. This will be 32 = 4 × side length. Therefore, the side length will be, 8 units.
 Step 2: Once the side length is known, the diagonal of the square can be calculated with the formula, Diagonal of square = a√2; where 'a' is the side length. Now, we can substitute this value in the formula, Diagonal of square = a√2 = 8 × √2 = 11.313 units.
How to find the Area of Square Using Diagonal?
The area of a square can be calculated if the length of diagonal is known. It is to be remembered that a square has all sides of equal length. We also know that the diagonal of a square formula, is d = a√2, where 'd' is the diagonal and 'a' is the side length of the square. Therefore, we can find the side length of the square using the diagonal length and this formula. After we know the side length of the square, we can easily find its area using the area of square formula, Area of square = a^{2}. Let us understand this with an example. If the diagonal of a square is 6 units, let us find the area of the square using the following steps:
 Step 1: The diagonal length = 6 units. So, let us substitute this value in the diagonal of a square formula, d = a√2, where 'd' is the diagonal and 'a' is the side length of the square. This will be, 6 = a√2. So, we get the value of a = 6/√2.
 Step 2: Now, we know the side length of the square which is 6/√2 units.
 Step 3: Let us find the area of square, Area of square = a^{2}. So, after substituting the value of a = 6/√2, we get Area = (6/√2)^{2}. This will be 36/2 = 18 square units.
How to find the Length of Diagonal of Square?
The length of diagonal of a square can be calculated if we know the side length of the square. The formula which is used to find the diagonal is, d = a√2, where 'd' is the diagonal and 'a' is the side length of the square. For example, if the side length of a square is 6 units, then we will substitute the value of a = 6 in the formula and its diagonal will be, d = 6√2 units = 8.48 units.
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