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 = l2.
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 4l, 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
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
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
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
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
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
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
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
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
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
Interested in learning more about the perimeter and area of geometric figures? Take a look at these pages:
