**The perimeter of a square is the total length measured across its boundaries.**

Let $x$ be** the length of each side** of the square, as shown in the figure below:

The perimeter is calculated using the formula:

$\textrm{Perimeter} = 4x$

The word perimeter is the combination of two Greek words, “Peri” meaning surrounding or enclosing of a surface, and “Meter” meaning measurement; so perimeter means **total measurement of boundaries of a surface**.

It is calculated by** adding all the sides of a given geometrical figure**, so if we add all the sides of a square, it will give us the perimeter of that square. This topic will help you understand the concept of the perimeter of a square and how to calculate it.

## What Is the Perimeter of a Square?

The perimeter of a square is** the total distance covered around its boundaries**. A square is a closed polygon with four equal sides, so if we multiply 4 with any of the sides, it will give us the perimeter of the square.

Sometimes, we are given the diagonal or the area of a square, and we are asked to calculate the perimeter. We will discuss how to find a perimeter in these scenarios.

The units of the perimeter are **the same** as the units of the length of the sides of a square and are given in centimeters, meters, inches, feet, etc.

## How To Find the Perimeter of a Square

To calculate the perimeter of a square, we have to **add all the sides of the square**. Consider the picture of a square given below.

If we add all the lengths, it will give us the perimeter of the square. This method is only applicable** if we are given the length of any side** of the square. In other cases, the perimeter can be calculated using:

- The diagonal of the square
- The area of the square

The given data will determine which method we have to use to calculate the perimeter of the square.

### Perimeter of a Square Using the Length of Its Sides

This method is used when **we are given the length of the sides of the square**. To calculate the perimeter using this method *we follow the steps below:*

- Write down the measurement of any one side of the square (for a square, all sides are equal).
- Multiply the length of the given side by “4”.
- Express the calculated perimeter in desired units.

### Perimeter of a Square Using the Diagonal of the Square

This method is used when** we are given the length of the diagonal** of the square.

To calculate the perimeter using this method, *we will follow the steps below:*

- Write down the measurement of the diagonal of the square.
- Calculate the length of the sides of the square by dividing the diagonal by $\sqrt{2}$. $Side = \dfrac{diagonal} {\sqrt{2}}$.
- The perimeter is calculated by multiplying the formula in step 2 by “4”. Perimeter $ = 4\times \dfrac{diagonal}{\sqrt{2}}$.

Perimeter $= (2\times 2) \dfrac{diagonal}{\sqrt{2}}$

Perimeter $= (2 \sqrt{2}) \times diagonal$

### Perimeter of a Square Using the Area

This method is used when **we are given the area of the square** and no data regarding the length of the side of the square is given. To calculate the perimeter using this method, *we will follow the steps listed below:*

- Write down the value of the area of the square.
- Calculate the length of one side of the square by using the following formula: Side $= \sqrt{area}$.
- The perimeter is calculated by multiplying the value of the side obtained in step 2 “4”. Perimeter $= 4\times \sqrt{area}$.

### Perimeter of a Square Formula

The perimeter of a square is very easy to derive. As we discussed earlier, the perimeter is calculated by **adding all the sides of the square**.

Perimeter of square = side + side + side + side

Side = x

The perimeter of a square is $= x+x+x+x$

Perimeter of square $= 4\times x$

## Real-life Applications of the Perimeter of a Square

The perimeter of a square can be used in **numerous real-life applications**. Various examples are given below:

- We can use the perimeter of a square to determine or estimate the length of a garden having a square shape.
- The perimeter formula is also helpful in designing a square table, cupboards and square swimming pool.
- It is also helpful in construction plans of square offices or a square boundary around a house.
- It is extremely helpful when farmers want to estimate the cost of fencing a square plot or a square farm.
- This formula will come in handy when building a square barn for horses. The perimeter of the square will help you in the construction of the barn.

### Example 1:

If the length of one side of the square is $7 \,cm$, what is the length of the remaining sides?

__Solution:__

We know that all the sides of a square are equal in length, so the length of the remaining three sides is also $7\,cm$ each.

### Example 2:

Calculate the perimeter of a square for the figure given below.

__Solution:__

We are given the length of one side of a square and we know that all the sides of a square are equal in length.

Perimeter of the square $= 4\times side$

Perimeter of the square $= 4\times 6$

Perimeter of the square $= 24\,cm$

### Example 3:

Suppose the perimeter of a square is $60\,cm$, what will be the length of all the sides of the square?

__Solution:__

We are given the perimeter of the square. We can calculate the length of a side of a square by using the perimeter formula

Perimeter of the square $= 4\times side$

$ 60 = 4\times side$

Side $= \dfrac{60}{4}$

Side $= \dfrac{60}{4}$

Side $= 15 \,cm$

We know that all the sides of the square are equal in length, so all the sides of the square are $15 \,cm$ each.

### Example 4:

If the length of one side of a square is $11 \,cm$, what will be the perimeter of the square?

__Solution:__

Perimeter of the square $= 4\times side$

Perimeter of the square $= 4\times 11$

Perimeter of the square $= 44\,cm$

### Example 5:

A square garden has an area of $49\, meter^{2}$. What will be the perimeter of the garden?

__Solution:__

As the garden has a square shape, we can calculate the length of any side of the garden by using the formula.

Side $= \sqrt{area}$

Side $= \sqrt{49}$

Side $= 7 \,m$

Perimeter of the square garden $= 4\times side$

Perimeter of the square garden $= 4 \times 7$

Perimeter of the square garden $= 28\, m$

### Example 6:

Nina is planning to design a square garden. If the length of the diagonal of the garden is $4\times \sqrt{2}\,meters$, what will be the perimeter of the garden?

__Solution:__

We are given the diagonal measurement of the garden.

Diagonal of the garden $= 4\times \sqrt{2}$ m

We can calculate the perimeter of the square garden by using the formula given below.

Perimeter of the garden $= (2\sqrt{2})\times \hspace{1mm} diagonal$

Perimeter of the garden $= (2\sqrt{2})\times 4 \sqrt{2}$

Perimeter of the garden $= 8\times 2$

Perimeter of the garden $= 16\,meters$

*Practice Questions*

1. If one side of the square is $10 \,cm$, what will be the length of the remaining sides and the value of the perimeter of the square?

2. If the perimeter of a square is $72\, cm$, what will be the length of the sides of the square?

3. Allan is designing a square table. Help Allan calculate the perimeter of the table using the data given below.

- The length of one side of the table is $20\,cm$.
- The diagonal of the table is $10\sqrt{2}\,cm$.
- The area of the table is $36\, cm^{2}$.

4. Nina is planning on constructing a square barn for her horses. Help Nina calculate the perimeter of the barn in centimeters using the data given below.

- The measurement of one side of the barn is $7\,meters$.
- The diagonal measurement of the barn is $5\sqrt{2}\,meters$.
- The area of the barn is $25\, meters^{2}$.

*Answer Key*

1. We are given the length of one side of the square and we know that all the sides of the square are equal so each side is = 10 cm.

Perimeter of the square $= 4\times side$

Perimeter of the square $= 4\times 10$

Perimeter of the square $= 40 \,cm$

2. We are given the perimeter of the square, so we have to find the length of one side of the square. Using the perimeter formula:

Perimeter of the square $= 4\times side$

$ 72 = 4\times side$

Side $= \dfrac{72}{4}$

Side $= \dfrac{60}{4}$

Side $= 18 \,cm$

As all the sides of the square are equal in length, the length of each side of the square is $= 18 \,cm$.

3.

- The length of one side of the square table is given, so we can calculate the perimeter by using the formula:

Perimeter of the table $= 4\times side$

Perimeter of the table $= 4\times 20$

Perimeter of the table $= 80\, cm$

- The length of the diagonal of the table $= 10\sqrt{2}\, cm$

We can calculate the perimeter of the table by using the formula:

Perimeter $= (2\sqrt{2})\times\hspace{1mm} diagonal$

Perimeter of the square table $= (2\sqrt{2})\times 10 \sqrt{2}$

Perimeter of the table $= (10\times 2) ( \sqrt{2}\times \sqrt{2})$

Perimeter of the table $= (20) ( 2)$

Perimeter of the table $= 40\, cm$

Area of the table = $36\, cm^{2}$

We can calculate the length of one side of the table by using the formula:

Side $= \sqrt{area}$

Side $= \sqrt{36}$

Side $= 6\, cm$

Perimeter of the table $= 4\times side$

Perimeter of the table $= 4 \times 6$

Perimeter of the table $= 24 \,cm$

4.

- One side of the barn $= 7m$

Perimeter of the barn $= 4\times side$

Perimeter of the barn $= 4\times 7$

Perimeter of the barn $= 28 \,meters$

But we are asked to calculate the perimeter in centimeters, so we have to convert the answer into centimeters.

Perimeter of the barn $= 28 \times 100 = 2800$ cm

- The length of the diagonal of the barn $= 5 \sqrt{2}\, meters$

Perimeter $= (2\sqrt{2})\times\hspace{1mm} diagonal$

Perimeter of the square table $= (2\sqrt{2})\times 5 \sqrt{2}$

Perimeter of the barn $= (5\times 2) ( \sqrt{2}\times \sqrt{2})$

Perimeter of the barn $= (10) ( 2)$

Perimeter of the barn $= 20\, m$

Perimeter of the barn $= 20 \times 100 = 2000\, cm$

- Area of the barn = $25 \,m^{2}$

We can calculate the length of one side of the table by using the formula

Side $= \sqrt{area}$

Side $= \sqrt{25}$

Side $= 5 m$

Perimeter of the barn $= 4\times side$

Perimeter of the barn $= 4 \times 5$

Perimeter of the barn $= 20 \; meters$

Perimeter of the barn $= 20 \times 100 = 2000 \;cm$