Area of Square Formula and Solved Examples

A 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. Squares are most commonly found in our surroundings in the form of carromboard, chessboards, a side of dice etc. We know that the area is in general 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 of the square.

In this article we will discuss about what is the area of square, how to find area of square using diagonal, etc and also understand the concept in detail.

What is the Area of Square?

A square is a 2 dimensional shape with four equal sides and four vertices. Squares can be found all around us in the form of chessboard, the clock, and a blackboard, etc. When we talk about perimeter and area of square, perimeter is found by taking the sum of all sides and area of square by squaring the measure of one side. Area of square is defined as the space coveredd by square in a 2D space or the xy plane. Now let us look into area of square formula.

Formula of area of square is defined as the product of the length of two of its sides. It is always measured in square units. Hence, Formula of area of square = Side × Side = ${S}^2$. The common units of the area of square are $\mathrm{m}^2$, inches ${ }^2, \mathrm{~cm}^2$, and $\mathrm{ft}^2$.

What is the formula of area of square?

From the above discussion, we now know that we can define the area of square as the product of the length of its sides.

Formula for area of square = $s \times s={s}^2$

where $s$ is the square side.

Area of square using the length of the diagonal:

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=\frac{d}{\sqrt{2 }} $.

Now, using the diagonal, Formula for area of square $=s^2=(\frac{d}{\sqrt{2 }})^2=\frac{d^2}{2}$. Hence, the area of square formula $=\frac{d^2}{2}$.

Surface Area of Square

Surface area of square is a measure of space or the area covered by square. The formula of area of square is applicable here. In other words, surface area of square is equal to the area of square. We can say that both terms refer to the area of square only. The unit is always in square units.

How to Find Area of Square?

We can find area of square depending on what values are given to us and what is missing. The values here include the sides of square or the diagonal. Some values might be given to us while some might be 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 formula example

Now let us look into some area of square formula examples.

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

Area of square is equal to $ side^2$
Hence, Area of 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 is equal to $ side^2$
Hence, the area of square $=2^2=2 \times 2=4 \mathrm{~cm}^2$

Area of Square using Diagonal

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

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 $=12 \mathrm{~cm}$
Hence, Area of square is equal to area of clipboard$=$ 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 square 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 floor $=$ length $\times$ breadth $=20 \mathrm{~m} \times 10 \mathrm{~m}=200 \mathrm{sq} . \mathrm{m}$
Side of square tile $=2 \mathrm{~m}$
Area of square tile $=$ side $\times$ side $=2 \mathrm{~m} \times 2 \mathrm{~m}=4 \mathrm{sq} \cdot \mathrm{m}$
No. of tiles required $= \frac{\text{ area of floor}}{\text{area of square tile}} =\frac{200}{4}=50$ 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 of square$=$ 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: What is area of square whose diagonal is 6 feet?
Solution:
The area of square when its diagonal is given is,

Formula for area of square $= \frac{\text{Diagonal }^2}{ 2}$.

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

Hence, area of square using diagonal $=\frac{(6 \times 6) }{2}=\frac{36}{ 2}=18$ square feet.

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

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