Area of Square

Area of Square

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Introduction to Square

A square is a closed two-dimensional shape with four equal sides and four equal angles.

Shape of a square is shown.

Squares are all around us.

Here are some commonly seen items which are square. 

Every day objects such as a chess board, a square wall clock, a square blackboard, a square biscuit are shown as examples.


What is a Square?

It is a quadrilateral in which:

  • the opposite sides are parallel.
  • all four sides are equal.
  • all angles measure 90\(^\circ\)

A square is a 2-dimensional shape with all 4 sides equal and all angles measuring 90 degrees.

Drag the vertex in the simulation below.

You can observe that all four sides in a square are equal and all the angles are 90\(^\circ\)


Definition of Area of a Square

The area of a square is the space occupied by the square.

Look at the green square shown below.

It has occupied 25 square units.

Area of a square - There is a green square within another square. The green square is further divided into 25 square units with numbers written inside each of the individual units. 

Therefore, the area of the square is 25 square units.

From the figure, we can observe that the length of each side is 5 units.

Therefore, the area of the square is the product of its sides.

Area of square = side \(\times\) side = 25 square units


Area of a Square Formula

The formula for the area of a square with side '\(s\)' is:

The area of a square formula is shown using a square with sides measuring s.

\(\begin{align} \text {Area of a Square} = s \!\times\! s \text{ units}^2 \end{align}\)

To understand this better, let us look at this solved example which uses the area of a square formula.

Find the area of a square of side 7 \(cm\).

A square with sides measuring 7 cm is shown. 

Area of a square of side 7 \(\text{cm}\) is 

\[\begin{align} 7 \text{ cm} \times 7 \text{ cm}= 49 \;\text{cm}^2 \end{align}\]

When the diagonal \(d\) of the square is known, the formula for finding the area of the square is: 

Area of a square is calculated from its diagonal d.

\(\begin{align} \text {Area of a Square} =  \frac{d^2}{2} \end{align}\)

You will understand these concepts better from the following solved examples for the area of a square formula.

 

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Area of a Square Sample Problems

Example 1

 

 

What is the area of a square swimming pool of side 8 \(m\)?

Examples for the area of a square formula - A square swimming pool has sides measuring 8 m.

Solution:

We know that the area of a square of side \(s\) is \(s^2\)

Therefore, the area of the swimming pool is

\[\begin{align} 8 \times 8 = 64 \: m^2\end{align}\]

\( \therefore\) Area of the swimming pool = 64 \(m^2\)
Example 2

 

 

The area of a square carrom board is 3600 \(cm^2\). What is the length of its sides? 

Examples for the area of a square formula - Area of a square carrom board is 3600 square cm.

Solution:

Area of a square of side \(s =s^2\)

Area of the square carrom board = 3600 \(cm^2\)

We know that 60 \(\times\) 60 gives 3600

\( \therefore\) Side of the carrom board = 60 \(cm\)
Example 3

 

 

Find the area of the floor of a square room which is made up of 100 tiles of side 5 \(cm\).

Examples for the area of a square formula - Find the area of the floor covered with 25 square tiles of sides 2m.

Solution:

Area of one tile 

\[\begin{align} &= 5 \text{ cm} \times 5 \text{ cm} \\ &= 25 \text{ cm}^2\end{align}\]

We know that there are 100 tiles on the floor of the room.

Thus, the area occupied by 100 tiles is the floor area

\[\begin{align} &= 100 \times 25 \text{ cm} \\ &= 2500 \text{ cm}^2 \end{align}\]

\(\therefore\) Area of the floor = 2500 \(cm^2\)
Example 4

 

 

Find the area of a square carpet whose diagonal is 4 feet. 

Examples for the area of a square formula - A square carpet has a diagonal measuring 16 ft.

Solution:

The area of a square when its diagonal is given is \(\begin{align}  \frac{d^2}{2} \end{align}\) 

Given, diagonal d = 4 ft

Area of the carpet

\[\begin{align}
&= \frac{4 \times 4}{2} \\\\
&=  \frac{16}{2} \\\\
&= 8\: \text{square feet}
\end{align}\]

\(\therefore\) Area of the carpet = 8 square feet
Example 5

 

 

A square wall of side 100 \(m\) needs to be painted.

What will be the cost of painting, if the painter charges Rs. 2 to paint an area of 1 \(m^2\)

Examples for the area of a square formula - A square wall of side 100 m has to be painted.

Solution:

Area of the square wall 

\[\begin{align} &= 100 \times 100 \\ &= 10,000 \: m^2 \end{align}\]

The cost of painting 1 \(m^2\) is Rs. 2

Thus, the cost to paint an area of 10,000 \(m^2\)

\(10,000 \times 2 = 20,000\)

 \(\therefore\) Cost of painting =  Rs. 20,000

Area of a Square Calculator

Use the area of a square calculator to find the area of any given square.

Enter the value in the space provided and click on Go to view the answer.

 
important notes to remember
Important Notes
  1. The area of a square of side \(s\) is \(s^2\)
  2. A square is a two-dimensional shape. Its unit is square units.
    Example: We can write it as \(cm^2\) or \(\text {square cm}\).
  3. A square has equal four sides and all angles measuring 90\(^\circ\)

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Practice Questions

Here are a few problems related to the area of a square formula.

Select/Type your answer and click the "Check Answer" button to see the result. 

 
 
 
 
 
 

Common Mistakes

  • A common mistake that we tend to do while calculating the area of a square is doubling the number.

    This is incorrect!

    Always remember that the area of a square is side \(\times\) side and not 2 \(\times\) side.
  • Units have to be mentioned when we represent the area. 

    The side of a square is one-dimensional and the area of a square is two-dimensional.

    Hence, the area of a square is always represented as square units.

    For example, a square of side 3 units will have an area of \(3 \times 3 = 9\) square units.
 
Challenge your math skills
Challenging Questions
  1. The diagonals of two squares are in the ratio 3 : 5

    What is the ratio of their areas?
  2. A wire of length 150 \(cm\) is bent to form a square and then to a circle.

    Which shape will have a more area?

Maths Olympiad Sample Papers

IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.

You can download the FREE grade-wise sample papers from below:

To know more about the Maths Olympiad you can click here


Frequently Asked Questions(FAQs) 

1. What is the formula for finding the area of a square?

The area of a square of side \(s\) units:

\(s \times s = \text{ units}^2\)

The area of a square when the diagonal \(d\) of the square is given:

\(\begin{align} \frac{d^2}{2} \end{align}\) square units

2. How do you calculate the area of a square?

Area of a square of side \(s\) units is

\(s \times s = \text{ units}^2\)

3. Give an example to show how the area of a square is calculated.

The area of a square of side \(s\) units is

\(s \times s\)

Example: Area of a square of side 5 cm

= \(5\text{ cm} \times 5\text{ cm} = 25\) \(cm^2\)

  
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