# Perimeter of a Square

## Perimeter of a Square – Introduction

In geometry, the perimeter of a shape is defined as the total length of its boundary. The perimeter of a square is determined by adding the length of all the sides. It is measured in linear units of measurement like centimeters, meters, inches, or feet.

## What is the Perimeter of a Square?

The perimeter of a square is the total length of all the sides of the square. Hence, we can find the perimeter of a square by adding all its four sides.

All the sides of a square are equal. So, the perimeter of a square is calculated by adding the side of a square four times.

## Perimeter of a Square Formula

To calculate the perimeter of a square, we use the following formula:

P $= 4 \times$ side $=$ 4a

## Derivation of Perimeter of Square Formula

Since the sides of a square are equal, that means, $a = b = c = d$

This means,

If we have a square with each side measuring 14 feet, then the perimeter of a square is given by $4 \times 14$ feet $= 56$ feet.

## How to Find the Perimeter of a Square?

We can find the perimeter of a square with side ‘a’ units by adding the lengths of all sides.

So, we add the side length ‘a’ four times.

Perimeter of a square $= (a + a + a + a)$ units

Perimeter of a square $= (4\times a)$ units

## Finding Perimeter of a Square when the Side is Unknown

Other than the side, we can find the perimeter of a square if we know its area. Let’s find out how.

Let’s consider a square with area A sq. units.

We know that area $= side^2$

So, using the area we can find the side of the square as $\sqrt{A}$.

And then we can use the measurement of the side to find the perimeter of the square.

Let us suppose we have to find the area of the square whose area is $36 \text{cm}^2$.

To find the perimeter of a square using the area, we can use the steps given below:

• Step 1: Find the side length using the area with the formula side $=$ A. In this example, side $= \sqrt{36} = 6$ cm
• Step 2: Apply the perimeter formula of the square, i.e., $4 \times$ side $= 4 \times 6 = 24$ cm

We can directly use the area to find the perimeter using this formula.

Perimeter of the square $= 4 \times (\sqrt{area}) = 4\sqrt{area}$ units.

## Solved Examples

1. What will be the perimeter of the square whose side is 10 cm?

Solution: Here, the side of the square $= 10$ cm

We know that perimeter of a square $= 4 \times$ side

So, the perimeter of the square with side $10$ cm $= 4 \times 10 = 40$ cm

2. If the area of the square is $144$ m2, what will be the perimeter?

Solution: Here, the area of square $= 144$ m2

We know that perimeter of the square whose area is given $= 4\sqrt{area}$

Perimeter of the square $= 4 \times \sqrt{144} = 4 \times 12$ m $= 48$ m

3. If the perimeter of a square is 56 cm, what will be the side of the square?

Solution: Here, the perimeter of square $= 56$ cm

We know that side of the square when perimeter is given = Perimeter$\div 4$So, the side of a square with perimeter 56 cm $= 56\div4 = 14$ cm

## Practice Problems

1

### Andy wants to frame 4 pictures. The pictures are square shaped, each side equal to 7 cm. Find the total length of the wooden sticks required to make the frames.

28 cm
49 cm
56 cm
112 cm
CorrectIncorrect
Side of 1 picture $= 7$ cm
Frame required for one picture $=$ Perimeter of the 1 picture $= 4 \times 7 = 28$ cm
Frame required for 4 pictures $= 4 \times 28 = 112$ cm
2

### Mary is building a square-shaped enclosure for her rabbit. Each side of the enclosure measures 15 m. Find the length of fence required, if Mary wants to fence from all 4 sides.

15 m
30 m
60 m
225 m
CorrectIncorrect
Side of the square enclosure $= 15$ m
Fence required for enclosure = Perimeter of the square enclosure $= 4 \times 15 = 60$ m
3

### The area of the square field is $81 \text{ft}^2$. What will be the cost of fencing the field if 1 foot costs $\$$2.5? \$$ 90$\$$22.5 \$$ 202.5
$\$$225 these CorrectIncorrect Correct answer is: \$$ 90 Area of the square field$= 81 \text{ft}^2$Perimeter of the square$= 4\sqrt{area} = 4 \times \sqrt{81} = 4 \times 9 \text{ft} = 36 \text{ft}\$