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# 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 (m^{2}).

** **

## 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) = √(2a^{2}) = 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) m^{2}

= 400 m^{2}

*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) cm^{2}

A = 625 cm^{2}

Therefore, the area of the square is 625 cm^{2}

*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) m^{2 }= 169 m^{2}

Now calculate the total cost of cementing by multiplying the area of the floor by the rate of cementing.

Cost = 169 m^{2} 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/m^{2}.

__Solution__

area = (150 x 150) = 22500 m^{2}

The cost of grassing = 22500 m^{2 }x $0.25/m^{2}

= $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 m^{2}.

__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 m^{2 }= [(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) m^{2}

= 324 m^{2}

*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) m^{2} = 3600 m^{2}

Area of a square tile = (2 x 2) m^{2 }= 4 m^{2}

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 m^{2})/ 4 m^{2}

= 900

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