## What is Diagonal?

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 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 as a regular quadrilateral and all four angles measure 900. 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 3600.

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 diagonal squares can be derived by using the Pythagoras theorem.

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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 diagonal of square.

The length of both the diagonals of a square are 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.

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

By using the Pythagoras theorem we can write,

d^{2 }= x^{2 }+ x^{2}

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

By simplifying this can be written as,

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

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

\[d=\sqrt{2}x\]

### 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 = 1/2 d^{2} square units

Here, d represents the length of the diagonal.

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} = 2x^{2}

x^{2} = 1/2 d^{2}

So, the area of square using diagonal formula = 1/2 d^{2} Square units.

### Solved Examples

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} = 1/2 d^{2}------ (1)

Here x^{2} = 36 square units

x = √36 square units

x = 6 units

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

\[d=\sqrt{2}x\]

\[d=\sqrt{2}\times6\]

d = 8.49 units

1. What is the Formula For Diagonal of Square?

Answer: A square consists of two diagonals where all the sides are of the same length which can be calculated by using the formula, d = √2 x, where x represents the side of the square.

2. Mention the Steps to Calculate Diagonal of a Square.

Answer: To calculate the length of a diagonal of a square using the formula,

**Step 1: **Check the length of a side of the square, and it is represented as ‘x’.

**Step 2:** Substitute the value of ‘x’ in the formula of the diagonal, d = √2 x .

**Step 3:** Write the value so obtained with the appropriate unit.