Diagonal of a Square Formula: Derivation, Solved Examples

Diagonal of a Square Formula: The line that extends from one corner of the square or rectangle to the opposite corner through the centre of the figure is known as the diagonal. Any square that has two diagonals is equal in length to each other. The diagonal of a square formula is used to calculate the square’s diagonals.

Diagonals are a line joining two non-adjacent vertices of a polygon, i.e., a diagonal joins two vertices of a polygon excluding the edges of the figure. The diagonal of a square is a line segment that connects any two non-adjacent vertices. A square has two diagonals equal in length and bisects each other at right angles. In this article, we will provide detailed information on the diagonal of a square formula. Scroll down to learn more!

Diagonal of Square Formula Derivation

In a square, the length of both the diagonals is the same. The length of a diagonal \(d\) of a square of side length \(x\) is calculated using the Pythagoras theorem. Observe the following square to see that the diagonal length is denoted by the letter \(d\) and the side length is denoted by \(x.\) Let us understand how to derive the formula to find the diagonal of a square.

The diagonal of a square divides the square into two equal triangles. Let us look at a diagonal of square formula derivation:
Let us consider the triangle \(PQR\) in the square. We know that all the angles in a square are \({90^{\rm{o}}};\) therefore, using the Pythagoras theorem, we can find the hypotenuse, which is \(d\) in this case.
We know that Pythagoras states that, in a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides.
Now, \({d^2} = {x^2} + {x^2}\)
\( \Rightarrow {d^2} = 2{x^2}\)
\( \Rightarrow {d^2} = \sqrt {2{x^2}} \)
\( \Rightarrow d = \sqrt 2 x\)
Therefore, the diagonal of a square formula is \(d = \sqrt 2 x.\)

Diagonal of a Square Formula Area

The magnitude of the measurement of the region enclosed by a closed plane figure is called its area.

The units of area are square centimetres (written as \({\rm{c}}{{\rm{m}}^2}\)), square meter (written as \({{\rm{m}}^2}\)), etc. The area of a square of side \(1\,{\rm{cm}}\) is \(1\,{\rm{c}}{{\rm{m}}^2}.\) The area of a square of side \(1\,{\rm{m}}\) is \(1\,{{\rm{m}}^2}.\)

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.
1. The length of the diagonals of a square is equal in measure.
2. The diagonal of a square are perpendicular bisectors of each other.
3. The diagonal of a square split the square into two congruent isosceles right-angled triangles.

The diagonal of a square formula is \(d = \sqrt 2 x;\) where \(d\) is the diagonal and \(x\) is the side of the square. The formula for the diagonal of a square has been derived using the Pythagoras theorem.

The diagonal split a square into the two isosceles right-angled triangles. The length of both the diagonals are congruent, and they bisect each other at right angles. And the area of a square using diagonal is \(\left[ {\frac{1}{2} \times {{\left( {{\rm{diagonal}}} \right)}^2}} \right]{\rm{sq}}{\rm{.units}}.\)

Diagonal of a Square Formula Area Derivation

The area of a square using diagonal length is as follows:

Here, diagonal of the square \( = \sqrt 2 x\,{\rm{units}}{\rm{.}}\) So, \(x = \frac{{{\rm{Diagonal}}}}{{\sqrt 2 }}.\)
We know that area of square \(x \times x = {x^2}\,{\rm{units}}{\rm{.}}\)
Now, \({x^2} = {\left( {\frac{{{\rm{Diagonal}}}}{{\sqrt 2 }}} \right)^2} = \left[ {\frac{1}{2} \times {{\left( {{\rm{diagonal}}} \right)}^2}} \right]{\rm{sq}}{\rm{.}}\,{\rm{units}}{\rm{.}}\)
Area of the square \( = \left[ {\frac{1}{2} \times {{\left( {{\rm{diagonal}}} \right)}^2}} \right]{\rm{sq}}{\rm{.}}\,{\rm{units}}{\rm{.}}\)

Half Diagonal of Square Formula

We know that diagonals of a square bisect one another at \({90^{\rm{o}}}.\) Then, \(\Delta AOB\) is the right-angled triangle by applying the Pythagoras theorem to \(\Delta AOB.\)
That is, \(A{B^2} = A{O^2} + B{O^2}\)
\( \Rightarrow {x^2} = {\left( {\frac{d}{2}} \right)^2} + {\left( {\frac{d}{2}} \right)^2}\)
\( \Rightarrow {x^2} = \frac{{{d^2}}}{4} + \frac{{{d^2}}}{4}\)
\( \Rightarrow {x^2} = 2 \times \frac{{{d^2}}}{4} = \frac{{{d^2}}}{2}\)
\( \Rightarrow x = \sqrt {\frac{{{d^2}}}{2}} \)
\( \Rightarrow x = \frac{d}{{\sqrt 2 }}\)
\( \Rightarrow x = \frac{d}{{\sqrt 2 }} \times \frac{{\sqrt 2 }}{{\sqrt 2 }}\)
\( \Rightarrow \frac{x}{{\sqrt 2 }} = \frac{d}{2}\)
Hence, the half diagonal of the square formula is \(\frac{d}{2} = \frac{x}{{\sqrt 2 }}.\)

Diagonal of a Rectangle Formula

Rectangle: A quadrilateral having opposite sides equal, and each angle measures \({90^{\rm{o}}}.\)
Consider a rectangle with length \( = l\) units and breadth \( = b\) units.

By Pythagoras theorem, length of diagonal \( = \sqrt {{l^2} + {b^2}} \,{\rm{units}}.\)

Also read,

Solved Examples on Diagonal of a Square Formula 

In Class 10, diagonal of square formula is an important concept, let us look at some of the solved questions:

Q.1. Compute the area of a square, the length of whose diagonal is \(6\sqrt 3 \,{\rm{m}}{\rm{.}}\)
Ans: We know that area of the square \( = \left[ {\frac{1}{2} \times {{\left( {{\rm{diagonal}}} \right)}^2}} \right]{\rm{sq}}{\rm{.units}}\)
\( = \left( {\frac{1}{2} \times {{\left( {6\sqrt 3 \,{\rm{m}}} \right)}^2}} \right){{\rm{m}}^2} = \left( {\frac{1}{2} \times 36 \times 3} \right){{\rm{m}}^2} = 54\,{{\rm{m}}^2}\)
Hence, the area of the square is \(54\,{{\rm{m}}^2}.\)

Q.2. Compute the length of each diagonal of a square with side \(12\,{\rm{cm}}{\rm{.}}\)
Ans: Given, the length of the side of the square is \(12\,{\rm{cm}}{\rm{.}}\)
Using the diagonal of square formula, the length of the diagonal, \(d\) is \(d = x\sqrt 2 \)
Now, the length of the diagonal is \(d = 12\sqrt 2 \,{\rm{cm}}.\)
Hence, the length of each diagonal of the given square is \(12\sqrt 2 \,{\rm{cm}}{\rm{.}}\)

Q.3. Suppose the length of the diagonal of a square is \(6\sqrt 2 \,{\rm{cm,}}\) then find the measure of the side of the square.
Ans: The diagonal of the given square \( = 6\sqrt 2 \,{\rm{cm}}{\rm{.}}\)
Let us assume the side length of the square to be \(x\)
In accordance with the diagonal of square formula, the length of the diagonal, \(d\) is: \(d = x\sqrt 2 \)
\( \Rightarrow 6\sqrt 2 = x\sqrt 2 \)
\( \Rightarrow x = 6\,{\rm{cm}}\)
Hence, the side length of the given square is \(6\,{\rm{cm}}{\rm{.}}\)

Q.4. Find the length of the diagonal of the rectangle whose dimensions are length \( = 15\,{\rm{cm}}\) and breadth \( = 8\,{\rm{cm}}.\)
Ans: Given, length \( = 15\,{\rm{cm}}\) and breadth \( = 8\,{\rm{cm}}\)
We know that length of diagonal \( = \sqrt {{l^2} + {b^2}} \,{\rm{units}}{\rm{.}}\)
Now, length of diagonal \( = \sqrt {{{15}^2} + {8^2}} = \sqrt {225 + 64} \)
\( = \sqrt {289} = 17\,{\rm{cm}}{\rm{.}}\)
Hence, the measure of the diagonal of the rectangle is \(17\,{\rm{cm}}{\rm{.}}\)

Q.5. The length of the diagonal of a square is \(18\sqrt 2 \,{\rm{units}}{\rm{.}}\) Find the length of the side of the square.
Ans: Given, the measure of diagonal of the given square \( = 18\sqrt 2 \,{\rm{cm}}.\)
Let us assume the side length of the square to be \(x\)
In accordance with the diagonal of square formula, the length of the diagonal, \(d\) is \(d = x\sqrt 2 \)
\( \Rightarrow 18\sqrt 2 = x\sqrt 2 \)
\( \Rightarrow x = 18\,{\rm{units}}\)
Hence, the side length of the given square is \(18\,{\rm{units}}{\rm{.}}\)

Summary

In this article, we learnt about diagonal of a square formula derivation, diagonal of a square formula area derivation, half diagonal of a square formula, and diagonal of a rectangle formula. The diagonal of a square formula is \(d = \sqrt 2 x.\). The area of a square using diagonal is \(\left[ {\frac{1}{2} \times {{\left( {{\rm{diagonal}}} \right)}^2}} \right]{\rm{sq}}{\rm{.units}}.\) Furthermore, the half diagonal of the square formula is \(\frac{d}{2} = \frac{x}{{\sqrt 2 }}.\)

In this article, we learnt how to find the length of the diagonal of a square with the given sides of the square by using the Pythagoras theorem application.

FAQs on Diagonal of a Square Formula

Let us look at some of the frequently asked questions about diagonal of a square formula:

Q.1. What is the diagonal formula of the square?
Ans: The diagonal of a square is a line segment that joins any two non-adjacent vertices. A square contains two diagonals that are equal in length and bisect with one another at right angles. The diagonal of the square formula is used to calculate the length of the diagonal of a square when its side length is known.
The diagonal of a square formula is \(d = \sqrt 2 x.\)
Here, \(d\) is the diagonal of the square, and \(x\) is the side of the square.

Q.2. What is the formula of diagonal of square and rectangle?
Ans: Diagonals of a square is equal in measure, and they bisect each other at the right angle.
The diagonal of a square formula is \(d = \sqrt 2 x.\)
Here, \(d\) is the diagonal of the square, and \(x\) is the side of the square.
The length of the diagonals of a rectangle is equal in measure.
The length of diagonal of a rectangle \( = \sqrt {{l^2} + {b^2}} \,{\rm{units}}\)
Here, \(l\) is the length of a rectangle, and \(b\) is the breadth of a rectangle.

Q.3. How to derive the diagonal of a square formula?
Ans: In a square, the length of the two diagonals is equal in measure. The length of a diagonal \(d\) of a square of side length \(x\) is calculated by using the Pythagoras theorem.
We know that Pythagoras declares that, in a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the remaining two sides.
So, \({d^2} = {x^2} + {x^2}\)
\( \Rightarrow {d^2} = 2{x^2}\)
\( \Rightarrow d = \sqrt {2{x^2}} \)
\( \Rightarrow d = \sqrt 2 x\)
Therefore, the diagonal of a square formula is \(d = \sqrt 2 x.\)

Q.4. How to calculate the diagonal of the square using the diagonal formula?
Ans: The diagonal of a square formula is \(d = \sqrt 2 x.\)
Example: If the length of the side of a square is \(8\,{\rm{cm}},\) then find the length of the diagonal of a square.
Solution: Given, the side of a square \( = 8\,{\rm{cm}}{\rm{.}}\)
We need to find the length of the diagonal of a square.
That is, the length of the diagonal of a square \( = \sqrt 2 \times 8\,{\rm{cm}}{\rm{.}}\)
Hence, the measure of the diagonal of a square is \(8\sqrt 2 \,{\rm{cm}}{\rm{.}}\)

Q.5. Is the diagonal of a square equal to its side?
Ans: No, the diagonal of a square is not equal to its side. Since all the square angles are equal to \({90^{\rm{o}}},\) the diagonal of a square becomes the hypotenuse of the triangle formed in the square.

Q.6. What do you understand by the diagonal of a square in Mathematics?
Ans: 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.
1. The length of the diagonals of a square is equal in measure.
2. The diagonal of a square are perpendicular bisectors of each other.
3. The diagonal of a square split the square into two congruent isosceles right-angled triangles.

We hope this detailed article on the diagonal of a square formula helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you.