Area of Square Formula and Solved Examples

The area is nothing but the space or the region covered by the object. While calculating the area of square, we have to observe only the length of its side. Since we are aware of the result that all sides of a square are equal; hence, area of square is equal to the square of the side.

What is Area of Square

Square is a closed two-dimensional shape with four equal sides and angles. The four sides form the four angles at the vertices or corners of the figure. Perimeter is found by taking the sum of all sides and area of square by squaring the measure of one side. Squares can be found all around us in the form of chessboard, the clock, and a blackboard, laptop, tablet, etc.

Area of Square Formula

Area of Square = Side × Side

Formula for area of square

Therefore, area of square = ${\text {Side} }^2 / 2$.
and the perimeter of square $=4 \times$ side units
Some important conversions that will be helpful in computations are listed below:

• $1 \mathrm{~m}=100 \mathrm{~cm}$
• $ 1 \mathrm{sq. m}=10,000 \space \mathrm{sq.cm}$
• $1 \mathrm{~km}=1000 \mathrm{~m}$
• $ 1 \mathrm{sq. km}=10,000 \space \mathrm{sq.m}$

We can use the diagonal of the square to find its area.

• Formula = ${\text {Diagonal} }^2 / 2$.

What is formula for area of square

The area of square is the product of the length of its sides:

$A=s \times s={s}^2$

where s is the square side.

From the following figure, 'd' is the diagonal and 's' represents the sides of the square.

Here the side of the square is ' $s$ ' and the diagonal of the square is ' $d$ '. We apply Pythagoras theorem,

$d^2=s^2+s^2$

$d^2=2 s^2$

$d=\sqrt{ 2s}$

$s=d / \sqrt{2 } $.

Now, using the diagonal, Area $=s^2=(d / \sqrt{2})^2=d^2 / 2$. Hence, the area of square $=d^2 / 2$.

Surface Area of Square

It is a measure of space or in simple terms the area covered by square. The formula of area of square is applicable here. In other words, surface area of square is equal to area of square. The unit is in square units always.

How to Find Area of Square

We can find area of square depending on what parameters are given to us and what is missing. So, let us see some situations when the perimeter of a square is given, when the sides of square are given, or when the diagonal is given.

Area of Square when the Perimeter of Square is Given

Example: Find the area of square park whose perimeter is 300 ft.
Solution:
Given: Perimeter of square park $=300 \mathrm{ft}$
We know that,
Perimeter of square $=4 \times$ side

$
\begin{aligned}
& \Rightarrow 4 \times \text { side }=300 \\
& \Rightarrow \text { side }=300 / 4 \\
& \Rightarrow \text { side }=75 \mathrm{ft}
\end{aligned}
$

Area of square $=$ side $^2$
Hence, Area of square park $=75^2=75 \times 75=5625 \mathrm{ft}^2$
Thus, the area of square park whose perimeter is 300 ft is $5625 \mathrm{ft}^2$

Area of Square When the Side of Square is Given

Example: Find the area of square whose side is 2 cm.
Solution:
Given: Side of square $=2 \mathrm{~cm}$
We know that,
Area of square $=$ Side $^2$
Hence, Area of square $=2^2=2 \times 2=4 \mathrm{~cm}^2$

Area of Square using Diagonal

Example: Find the area of square whose diagonal is 10 cm.
Solution:
Given: Diagonal of square $=10 \mathrm{~cm}$
We know that,
Area of square using diagonal $=d^2 / 2$
Hence, Area of square using diagonal $=(10 \times 10) / 2=50 \mathrm{~cm}^2$

Tips to Find Area of Square

We should take care of the following points while we calculate the area of a square.

• A common mistake that we tend to make while calculating the area of the square is that we double the number which is wrong. We should see that the area is side × side and not 2 × side.
• When we represent the area of square, we should not forget to write its unit.

Area of a Square Sample Problems

Example 1: Find the area of square clipboard whose side measures 12 cm .
Solution:
Side of the clipboard that is in shape of square $=12 \mathrm{~cm}$
Area of square $=$ side $\times$ side

$
\begin{aligned}
& =12 \mathrm{~cm} \times 12 \mathrm{~cm} \\
& =144 \mathrm{sq} \cdot \mathrm{~cm}
\end{aligned}
$

Example 2: The side of square wall is 70 m. What is the cost of painting it at the rate of Rs. 3 per sq. m?

Solution:
Side of the wall $=70 \mathrm{~m}$
Area of the wall $=$ side $\times$ side $=70 \mathrm{~m} \times 70 \mathrm{~m}=4900 \mathrm{sq} . \mathrm{m}$
For 1 sq. m, the cost of painting = Rs. 3
Thus, for 5,625 sq. $m$, the cost of painting $=$ Rs. $3 \times 4900=$ Rs 14700 .

Example 3: A courtyard's floor which is 20 m long and 10 m wide is to be covered by square tiles. The side of each tile is 2 m. Find the number of tiles required to cover the floor.

Solution:
Length of the floor $=20 \mathrm{~m}$
The breadth of the floor $=10 \mathrm{~m}$
Area of the floor $=$ length $\times$ breadth $=20 \mathrm{~m} \times 10 \mathrm{~m}=200 \mathrm{sq} . \mathrm{m}$
Side of one tile $=2 \mathrm{~m}$
Area of one tile $=$ side $\times$ side $=2 \mathrm{~m} \times 2 \mathrm{~m}=4 \mathrm{sq} \cdot \mathrm{m}$
No. of tiles required $=$ area of floor/area of a tile $=200 / 4=50$ tiles.

Example 4: The area of a square-shaped carrom board is $360 \mathrm{~cm}^2$. What is the length of its side?
Solution:
Area of the square carrom board $=360 \mathrm{~cm}^2$.

We know that Area $=$ side $\times$ side $=$ side $^2$.
So, side $=\sqrt{ Area}$ $=\sqrt{360}=18.9 \mathrm{~cm}$.

Therefore, the side of the carrom board is 18.9 cm.

Example 5: Find the area of square whose diagonal is 6 feet.
Solution:
The area of square when its diagonal is given is,

Area of square $=$ Diagonal $^2 / 2$.

Given, diagonal $(\mathrm{d})=6 \mathrm{ft}$.

Area of square $=(6 \times 6) / 2=36 / 2=18$ square feet.

Therefore, the area of square is equal to 18 square feet.

Summary

In this article, we have learned about the area of square. We have also learnt about various ways of finding it with the known parameters. This figure is most commonly observed in our daily lives such as board, chessboard and other indoor games. We have also learnt about formula of area of square, surface area of square, and area of square using diagonals. Perimeter and area of square is used almost daily in our lives.