## Diagonal of Square

The **diagonal of a square** is a line segment that joins any two non-adjacent 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 right-angled 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 right-angled 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

## 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 the 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 non-adjacent 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 right-angled triangles.

### What is the Formula for the Diagonal of a 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 Calculate the Diagonal of the 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 of a square formula can be derived by using the Pythagoras theorem.

- Step 1: Draw the diagonals of a square.
- Step 2: Two right-angled 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 the 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.

## Diagonal Of A Square Formula

In geometry, a square is a regular quadrilateral that has four equal sides and four angles which are 90∘90∘ in measurement. Thus the sum of all the internal angles of a square is equal to 360∘360∘.

**Diagonal of a square-**

A square can have two diagonals. Each of the diagonal can be formed by joining the diagonally opposite vertices of a square. The properties of diagonals are as follows-

- Both the diagonals are congruent (same length).
- Both the diagonals bisect each other, i.e. the point of joining of the two diagonals is the midpoint of both the diagonals.
- A diagonal divides a square into two isosceles right-angled triangles.

**Length of the Diagonal-**

Consider a square ABCD having sides equal to ‘a’ cm.

Let AD and BC be the two diagonals of a square.

## The Properties of a Square

## Diagonal of a Square Formula: Derivation, Area, Half Diagonal

**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 are equal in length to each other. The diagonal formula is used to calculate the polygon 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 that are equal in length and bisect each other at right angles. The diagonal of a square formula is used to calculate the length of the diagonal of a square when its side length is known.

## Diagonal of Square Formula Derivation

In a square, the length of both the diagonals is the same. The length of a diagonal dd 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 dd and the side length is denoted by x. Let us understand how to derive the formula to find the diagonal of a square.

## Diagonal of a Square Formula Area

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

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.

## Diagonal of a Square Formula Area Derivation

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

## Half Diagonal of Square Formula

## Diagonal of a Rectangle Formula

## Solved Examples – Diagonal of a Square Formula

### Frequently Asked Questions – Diagonal of a Square Formula

** Q.1. What is the diagonal formula of the square?** 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.

Ans:

** Q.6. What do you understand by the diagonal of a square in Mathematics?**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.

Ans:

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.

## What is Diagonal?

We have studied what is a polygon. To recall, a two-dimensional closed figure bounded with three or more than three straight lines is called a polygon. Triangles, square, rectangle, pentagon, hexagon, are some examples of polygons.

The segments are referred to as the sides of the polygon. The points at which the segments meet are called vertices. Segments that share a vertex are called adjacent sides. A segment whose endpoints are nonadjacent vertices is called a diagonal.

A diagonal of a polygon is a line from a vertex to a non-adjacent vertex. So a triangle has no diagonals. You cannot draw a line from one interior angle to any other interior angle that is not also a side of the triangle. A quadrilateral, the next-simplest, has two diagonals. A pentagon has five diagonals. The shapes with a diagonal drawn on them are given below:

We have a number of diagonals formula of a polygon which help us to know the number of diagonals in a polygon that is

Where the number of sides of a polygon is n

From this formula, we can easily calculate the number of diagonals in a polygon.

**The Given Table Shows the Number of Diagonals in Different Polygons**

Let us study diagonal formula for different shapes

**Diagonal of Square Formula**

Where s is the side of a cube.

**Diagonal of Cuboid Formula**

## FAQs (Frequently Asked Questions)

**1. State Diagonals Examples in Real Life.**

Answer: Some of the real-life examples of diagonals are:

- Diagonals in squares and rectangles add strength to construction, whether for a house wall, bridge, or tall building.
- You may see diagonal wires used to keep bridges steady.
- When houses are being built, look for diagonal braces that hold the walls straight and true.
- Bookshelves and scaffolding are braced with diagonals.
- For a catcher in softball or baseball to throw out a runner at second base, the catcher throws along a diagonal from home plate to second.
- The phone or computer screen is measured along its diagonal.

**2. What is the Formula to Find the Number of Diagonals in any Polygon?**

Answer: Formula for the number of diagonals

As we know that, the number of diagonals from a single vertex is three less than the number of vertices or sides, or (n-3).

Consider there are n number of sides and n vertices, which gives us n(n-3) diagonals. Each diagonal has two ends, so this would count each diagonal twice. So we divide by 2, for the final formula:

Number of diagonals = n(n−3)/ 2

Where n is the number of sides.

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