The perimeter of a square represents the sum of the lengths of all its sides. On the other hand, the area of a square represents the space occupied by the square in two-dimensional space. **We can calculate the perimeter of a square using the formula p = 4l and we can calculate the area of a square using the formula A = l^{2}.**

In this article, we will learn about the perimeter and area of a square in detail. We will learn about their formulas and use them to solve some practice problems.

## How to find the perimeter of a square?

To find the perimeter of a square, we can add the lengths of its four sides. Since all sides of a square are the same length, **the formula for calculating the perimeter of a square is 4 multiplied by the length of one side:**

Perimeter = 4 × side $latex p=4l$ |

### Proof of the formula for the perimeter of a square

The formula for the perimeter of the square is given by:

Perimeter = Sum of all four sides

Perimeter = side + side + side + side

Perimeter = 4 × side

Therefore, the perimeter of the square is equal to 4*l*, where *l* is equal to the length of one side of the square.

## How to find the area of a square?

We can find the area of a square by squaring the length of one of the square’s sides. Therefore, we have the following formula:

$latex A={{l}^2}$ |

where,

*A*is the area of the square*l*is the length of one of the sides of the square

### Proof of the formula for the area of a square

To prove the formula for the area of a square, we are going to find the area of a square that has sides with a length of 5 cm, as shown in the diagram below.

The square is drawn on a 1 cm × 1 cm grid. Therefore, the square we drew covers 25 of the small squares.

This means that the area of the square is 25 cm², which can be written as 5 cm × 5 cm, that is, we have side × side. Thus, we have that the area of the square is:

Area = Side × Side Area = Side² $latex A={{l}^2}$ |

## Perimeter and area of a square – Examples with answers

**EXAMPLE 1**

Find the perimeter of a square that has sides with a length of 8 inches.

##### Solution

We use the formula for the perimeter of a square with the length *l*=8:

$latex p=4l$

$latex p=4(8)$

$latex p=32$

Therefore, the perimeter of the square is 32 in.

**EXAMPLE **2

**EXAMPLE**

What is the area of a square that has sides with a length of 12 feet?

##### Solution

We use the formula for the area of a square with length *l*=12 ft.

$latex A={{l}^2}$

$latex A={{12}^2}$

$latex A=144$

Therefore, the area of the square is 144 ft².

**EXAMPLE **3

**EXAMPLE**

If a square has sides with a length of 12 inches, what is its perimeter?

##### Solution

By applying the perimeter formula with the length *l*=12, we have:

$latex p=4l$

$latex p=4(12)$

$latex p=48$

The perimeter of the square is 48 in.

**EXAMPLE **4

**EXAMPLE**

Find the area of a square that has sides with a length of 15 yards.

##### Solution

The length of one side of the square is 15 yd, so we use the area formula with this value:

$latex A={{l}^2}$

$latex A={{15}^2}$

$latex A=225$

Therefore, the area of the square is 225 yd².

**EXAMPLE **5

**EXAMPLE**

Find the perimeter of a square that has sides with a length of 25 inches.

##### Solution

We use the value *l*=25 in the formula for the perimeter, and we have:

$latex p=4l$

$latex p=4(25)$

$latex p=100$

Therefore, the perimeter of the square is 100 in.

**EXAMPLE **6

**EXAMPLE**

A square wall has sides with a length of 6 feet. What is the cost of painting the wall if we have a rate of 0.5 USD per square foot?

##### Solution

First, we have to find the area of the wall. Therefore, we use the formula $latex A={{l}^2}$ with the length *l*=6:

$latex A={{l}^2}$

$latex A={{6}^2}$

$latex A=36$

The area of the wall is 36 square feet. If the rate is 0.5 dollars per square foot, the cost will be:

$latex 36\times 0.5=18$ USD

**EXAMPLE **7

**EXAMPLE**

Determine the length of the sides of a square that has a perimeter of 44 feet.

##### Solution

Here, we have the perimeter and we want to determine the length of one of the sides. Therefore, we use the formula for the perimeter and solve for *l*:

$latex p=4l$

$latex 44=4l$

$latex l=11$

Therefore, the length of each side of the square is 11 ft.

**EXAMPLE **8

**EXAMPLE**

Find the length of one of the sides of a square that has an area of 121 in².

##### Solution

Here, we know the area and we want to find the length of one of the sides of the square. Therefore, we use the area formula and solve for *l*:

$latex A={{l}^2}$

$latex 121={{l}^2}$

$latex l=\sqrt{121}$

$latex l=11$

Therefore, the length of one side of the square is 11 inches.

**EXAMPLE **9

**EXAMPLE**

What is the length of the sides of a square that has a perimeter of 60 inches?

##### Solution

The perimeter of the square is 60 in. Therefore, we use the perimeter formula with the value *p*=60 and solve for it to find the length of the sides:

$latex p=4l$

$latex 60=4l$

$latex l=15$

The length of the sides of the square is equal to 15 in.

**EXAMPLE **10

**EXAMPLE**

A square floor that has sides with a length of 40 feet is to be covered with ceramics. If each tile has sides that are 2 feet long, how many tiles are needed to cover the floor?

##### Solution

We have to find both the area of the floor and the area of each tile. Thus, the area of the floor is:

$latex A_{p}={{l_{p}}^2}$

$latex A_{p}={{40}^2}$

$latex A_{p}=1600$

The floor area is 1600 ft² and the area of each tile is:

$latex A_{c}={{l_{c}}^2}$

$latex A_{c}={{2}^2}$

$latex A_{c}=4$

The area of each tile is 4 ft². Therefore, we need:

$latex \frac{A_{p}}{A_{c}}=\frac{1600}{4}=400$ tiles

## Perimeter and area of a square – Practice problems

## See also

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