# Area of Squares – Explanation & Examples

As explained in the previous article about quadrilaterals, a square is a regular polygon with four equal sides and four right angles.

Now that you are already acquainted with the term area. In this article, you will learn about the area of a square and how to find the area using the area of a square formula.

## How to Find the Area of a Square?

In the square ABCD shown below, the lengths AB = BD = DC = AC = a

The area of a square is, therefore, the region occupied inside the sides of a square. The measurement of the area is done in square units, with the standard unit being square meters (m2).

## Area of a Square Formula

The area of a square can be calculated by drawing a square on a graph paper having 1 cm × 1 cm squares. After drawing the square, you can count the total number of complete squares and incomplete squares.

The area of the square is then approximated as;

Area = Number of complete squares + ½ (number of incomplete squares)

This method of finding an area of a square is just an approximation and cannot be used where accurate figures are required.

For this reason, let’s look at the most accurate formula for calculating the area of a square.

For a square of side length, a, the area of a square states that:

Area of a square = side × side

A = (a × a) sq. unit

Therefore,

Area of a square = a² square units

Alternatively, we can calculate the area of a square as:

Area of a square = a × a = (P/4) ² sq. units

where P = perimeter of a square.

Additionally, the area of a square can be calculated using its diagonal as;

Area of a square = 1/2 × (diagonal)² sq. units

But the diagonal of a square is calculated by Pythagorean theorem as,

Diagonal = √ (a² + a²) = √(2a2) = a√2

Where a = side length of a square.

Let’s work out a few example problems about the area of a square.

Example 1

Find the area of a square of side 20 m.

Solution

Area of a square = (a x a) Sq. unit

By substitution,

= (20 × 20) m2
= 400 m2

Example 2

Find the area of a square whose perimeter is 100 cm.

Solution

Perimeter of square = 100 cm

The perimeter of square = 4 × side

Therefore, 4 × side = 100 cm

Divide both sides by 4.

side = a = (100/4) cm = 25 cm

Now substitute a = 25 in the area of a square formula.

Area of a square = (25 x 25) cm2

A = 625 cm2

Therefore, the area of the square is 625 cm2

Example 3

Find the cost of cementing a square floor of side 13 m if the rate of cementing is $10 per m². Solution First, calculate the area of the square floor. Area of a square = (a x a) Sq. unit = (13 x 13) m2 = 169 m2 Now calculate the total cost of cementing by multiplying the area of the floor by the rate of cementing. Cost = 169 m2 x$10 per m².

= $1690 Example 4 The length of a square football pitch is 150 m. Calculate the cost of grassing the pitch if the rate is$0.25/m2.

Solution

area = (150 x 150) = 22500 m2

The cost of grassing = 22500 m2 x $0.25/m2 =$5,625

Example 5

Find the area of a square lawn rounded by a path of 2 wide. Take the area of the path to be 160 m2.

Solution

Let the sides of the lawn be x, and the side of the lawn plus the path be x + 4.

Therefore,

The area of the path = (area of the lawn including the path) – (area of the lawn)

160 m2 = [(x * 4) (x + 4)] – (x * x)

160 = x² + 8x + 16 – x²

Simplify

160 = 8x + 16

Subtract 16 on both sides,

144 = 8x

Divide both sides by 8.

144/8 = x

18 = x

Therefore, the area of the lawn = (18 x 18) m2

= 324 m2

Example 6

A square courtyard’s floor, which 60 m, is to be covered by square tiles. Find the total number of tiles needed to completely cover the floor if the length of a tile is 2 m.

Solution

Calculate the area of both the square courtyard’s floor and the square tile.

Area of the courtyard’s floor = (60 x 60) m2 = 3600 m2

Area of a square tile = (2 x 2) m2 = 4 m2

To find the number of tiles needed to cover the courtyard’s floor, divide the area of the courtyard’s floor by the area of a tile.

Number of tiles = (3600 m2)/ 4 m2

= 900

Therefore, 900 tiles are needed to cover the courtyard’s floor completely.

### Practice Questions

1. What is the area of a square with a side that is $15$ ft long?

2. What is the area of a square with a side that is $4\sqrt{2}$ meters long?

3. What is the area of a square that has a perimeter of $200$ cm?

4. If it costs $1.20$ dollars per squared feet for wallpapers, how much would it cost to redo the wallpaper of a square wall with a side that is $40$ feet long?

5. A bathroom floor is to be covered by square tiles. How many tiles are needed to completely cover the floor if the length of a tile is $1.5$ m and the floor is $45$ m long on each side?