area of square assignment expert

How to Find the Area of a Square in Python?

Recently, someone asked me in a one-to-one session to “write a program to find the area of a square in Python.” I explained it in detail. In this tutorial, I will show you how to find the area of a square in Python with examples.

To find the area of a square in Python using basic arithmetic, you simply need to square the length of one side. For example, if the side length is 5 units, you can calculate the area by defining the side length and then using the exponentiation operator ( ** ) to square it:

This will output: “The area of the square is: 25”.

Table of Contents

What is the area of the square in Python?

The area of a square is the amount of space enclosed within its four equal sides. The formula to calculate the area is simple:

Where “side” is the length of one side of the square.

Find the Area of the Square in Python

Now, let me show you how to find the area of the square in Python using different methods.

Method 1: Using Basic Arithmetic

Let’s start with a basic example: In Python, we use simple arithmetic operations to calculate the area of a square.

Below is an example.

Example 1: Basic Arithmetic

In this example, we define the length of the side of the square as 5 units. We then use the exponentiation operator ( ** ) to square the side length, and store the result in the variable  area . Finally, we print the area.

I executed the above Python code, and you can see the output in the screenshot below:

Read Write a Python Program to Add N Numbers Accepted from the User

Method 2: Using Functions

Using functions is a good practice as it makes the code reusable and modular. Let’s create a function to calculate the area of a square in Python.

Here is the Python program to find the area of the square using a function.

Example 2: Using Functions

In this example, we define a function  calculate_area  that takes one parameter,  side . The function returns the area by squaring the side length. We then call this function with a side length of 7 units and print the result.

Here is the output in the screenshot below; you will also receive the exact output:

Read Write a Program to Check Whether a Number is Prime or not in Python

Method 3: Using User Input

To make the program more interactive, we can take the side length as input from the user. Now, let me write a program to find the area of a square in Python that will take user input.

Example 3: Using User Input

Here, we use the  input  function to take the side length as input from the user. The  float  function converts the input to a floating-point number to handle decimal values. We then call the  calculate_area  function and print the result.

Read Write a Python Function to Find the Max of Three Numbers

Method 4: Using Classes

For more advanced users, using classes can be a good way to encapsulate the properties and methods related to a square. Let me explain to you with a complete example.

Example 4: Using Classes

Here is the complete Python code.

In this example, we define a class  Square  with an initializer method  __init__  that sets the side length. The class also has a method  calculate_area  that returns the area of the square. We create an instance of the  Square  class with a side length of 4 units and call the  calculate_area  method to get the area.

Here is the output, you can see in the screenshot below:

In this tutorial, we covered different methods for  calculating the area of a square in Python . We started with basic arithmetic and moved on to using functions, user input, and classes. However, I always recommend using the function method .

You may also like the following tutorials:

• Write a Program to Calculate Simple Interest in Python
• Write a Python Program to Remove Duplicates From a Dictionary
• Write a Python Program to Check Whether a List Contains a Sublist
• Write a Program to Find a Perfect Number in Python

I am Bijay Kumar, a Microsoft MVP in SharePoint. Apart from SharePoint, I started working on Python, Machine learning, and artificial intelligence for the last 5 years. During this time I got expertise in various Python libraries also like Tkinter, Pandas, NumPy, Turtle, Django, Matplotlib, Tensorflow, Scipy, Scikit-Learn, etc… for various clients in the United States, Canada, the United Kingdom, Australia, New Zealand, etc. Check out my profile .

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Program for Area Of Square

A square is a flat shape, in one plane, defined by four points at the four corners. A square has four sides all of equal length, and four corners, all right angles (90 degree angles). A square is a kind of rectangle.   

Examples :    

Output :    

Time complexity : O(1)  Auxiliary Space : O(1)

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Area of Square in Python

Python program to get side of square and find area of square

Sample Input 1:

Sample Output 1:

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Answer to Question #236438 in Python for vijay

ou are given a side of a square as input. Write a program to find the perimeter and area of the square.

The input is an integer representing the length of the side of the square.

The first line of the output should contain area of the square, and the second line of the output should contain the perimeter of the square as per the format shown in the sample output.

Explanation

Given the length of the side is 2.

As the area of square is side*side and the perimeter of the square is 4*side.

Then output should be

Sample Input 1

Sample Output 1

Sample Input 2

Sample Output 2

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Python Program to calculate area of a square

In this tutorial, we will learn how to do a simple program of calculating the area of a square in python. The area is defined as the total space a shape occupies. It is measured in square units like cm², m², km² depending on the unit of the dimensions. The formula for calculating the area of a square is given as

The formula for Area of Square

Area of a square= Side x Side

Area= Side²

For example, following as input and give the output accordingly.

Output- 6.25

Here are two simple methods for calculating and printing the area of a square where the measurement of a side is given by the user.

• Using multiplication operator (*)
• Using pow() function

Approach 1: Using multiplication operator (*)

Given below is a simple program for calculating area using the multiplication operator (*). The input is taken as float and the area is calculated up to 4 decimal places. We will use the "%.4f" specifier for getting 4 digits after the decimal point. In "%.4f" the number after the dot is used to indicate the decimal places and f specifies float.

Step 1 - Take input of side from user

Step 2 - Calculate area

Step 3 - Print area using "%.4f"

Python Program

Look at the program to understand the implementation of the above-mentioned approach.

Enter side of square3.2 Area of square= 10.2400

Approach 2: Using pow() function

pow() is a predefined math function in python which returns the value of x to the power y. To know more about pow() and other built-in math functions. I advise you to read the article on Python math function .

Step 1 - Define a function area_square() to calculate area Take input of side from user

Step 2 - Call pow() and set parameters as n ,2 to calculate the area

Step 3 - Take input from the user

Step 4 - Call area_square() and pass input as a parameter

Step 5- Print the area

Enter side of square2.4 Area of square= 5.7600

In this tutorial, we learned how to calculate the area of a square using 2 approaches. One, by using simple statements for multiplication and printing the output. Two, using a predefined math function called pow(). You can also define a function to calculate area by simply using the code from the first approach.

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Area of a Square Calculator

Table of contents

If you forgot how to find the area of a square, you're in the right place - this simple area of a square calculator is the answer to your problems. Whether you want to find the area knowing the square side or you need to calculate the side from a given area, this tool lends a helping hand. Read on and refresh your memory to find out what is the area of a square and to learn the formula behind the calculator. If you also need to calculate the diagonal of a square, check out this square calculator .

Formulas for the area of a square

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

where a a a is a square side.

Other formulas also exist. Depending on which parameter is given, you can use the following equations:

• A = d 2 / 2 A = d^2 / 2 A = d 2 /2 if you know the diagonal ;
• A = P 2 / 16 A= P^2 / 16 A = P 2 /16 if the perimeter is given (you can learn how to find P P P in every possible way with our perimeter of a square calculator );
• A = 2 × R 2 A= 2 \times R^2 A = 2 × R 2 knowing circumradius R R R ; and
• A = 4 × r 2 A= 4 \times r^2 A = 4 × r 2 in terms of the inradius r r r .

What is the area of a square?

The area of a square is the number of square units needed to completely fill a square. To understand that definition, let's have a look at this picture of a chessboard:

The board has a squared shape, with its side divided into eight parts, in total, it consists of 64 small squares. Assume that one small square has a side length equal to 1   i n 1\ \mathrm{in} 1   in . The area of the square may be understood as the amount of paint necessary to cover the surface. So, from the formula for the area of a square, we know that A = a 2 = 1   i n 2 A= a^2 = 1\ \mathrm{in^2} A = a 2 = 1   i n 2 , and it's our unit of area in the chessboard case (amount of paint). The area of a 2 x 2 piece of the chessboard is then equal to 4 squares - so it's 4   i n 2 4\ \mathrm{in^2} 4   i n 2 , and we need to use 4 times more "paint". Full chessboard area equals 64   i n 2 64\ \mathrm{in^2} 64   i n 2 : 8   i n × 8   i n 8\ \mathrm{in} \times 8\ \mathrm{in} 8   in × 8   in from the formula, or it's just 64 small squares with 1   i n 2 1\ \mathrm{in^2} 1   i n 2 area - so we need 64 times more "paint" than for one single square.

You may also be interested in checking out the area of the largest square inscribed in a circumference with our square in a circle calculator !

How to use the area of a square calculator

Let's give the area of a square calculator a try!

• Find out the given value . In our example, assume we know the side and want to calculate the area.
• Type it into the proper box . Enter the value, e.g., 11 11 11 inches, into the side box.
• The area appears! It's 121   i n 2 121\ \mathrm{in^2} 121   i n 2 . If you are interested in how many square feet it is, change the unit by clicking on the unit name.

How do I find the area of a square given perimeter?

If you know the perimeter of a square and want to determine its area, you need to:

• Divide the perimeter by 4.
• The result is the side of the square.
• Multiply the side by itself.
• The result is the area of your square.

How do I find the diagonal of a square given area?

To compute the length of a diagonal of a square given its area, follow these steps:

Multiply the area by 2 .

Take the square root of the result of step 1 .

That's it! The result is the diagonal of your square. Congrats!

The formula we used here is:

diagonal = √(2 × area)

What is the area of a square with diagonal 10?

The answer is 50 . This is because the formula linking the area of a square with its diagonal is:

area = diagonal² / 2

Hence, plugging in diagonal = 10 , we obtain:

area = 100 / 2 = 50

What is the area of a square with perimeter 52?

The answer is 169 . To arrive at this result, observe that the perimeter is equal to 52 . This means that the side of the square equals:

side = perimeter /4 = 52 / 4 = 13

Hence, the area is:

area = 13² = 169 .

© Omni Calculator

Area of a Square Calculator

Use this calculator to easily calculate the area of a square given the length of its side in any metric: mm, cm, meters, km, inches, feet, yards, miles...

    Calculation results

Related calculators

• Area of a square formula
• How to calculate the area of a square?
• Example: find the area of a square

    Area of a square formula

The formula for the area of a square is side 2 , as seen in the figure below:

The solution to the equation is straightforward multiplication and this is the formula used in our area of a square online calculator. The result will be in whatever metric you did the measurement in, but squared: square mm, square cm, square dm, square meters or square inches, square feet, square yards, square miles etc.

    How to calculate the area of a square?

The area of a square is one of the easiest to calculate in that it requires just one measurement of the square to be known - its side. You then multiply it by itself to get to the area, so the formula used in this area of a square calculator is just as simple.

The simplicity of the square is why it is usually one of the first shapes that geometry students become familiar with. In real life measurements, like in construction, engineering, landscaping, etc. we rarely deal with square areas and surfaces - they are more often rectangular in shape. Still it is a good teaching device as it contains the rules for solving many other shapes.

This is because a square can be interpreted as a special case of a rhombus (equal sides and opposite equal angles), a kite (two pairs of adjacent equal angles), a trapezoid (one pair of opposite sides are parallel), a parallelogram (all opposite sides are parallel), and, of course, a rectangle, in which the opposite sides are equal in length and all of its angles are 90°. Therefore, a square combines the properties of all of these shapes: diagonals bisect at 90°, diagonals bisect the square angles, diagonals are equal, the sides are equal, opposite sides are equal, all angles are equal (90°). Learning to calculate the area of a square is thus a precursor for learning how to calculate the areas of more advanced shapes.

    Example: find the area of a square

The only measurement needed to find the area of a square figure is its side. Since all sides are equal it does not matter which side is measured. Then simply multiply the measurement by itself to get the area.

For example, if the side of a square pool is 10 yards, then the pool area is 10 x 10 = 100 square yards. If one side of a square room is 12 feet, then its area is 12 x 12 = 144 square feet.

Cite this calculator & page

If you'd like to cite this online calculator resource and information as provided on the page, you can use the following citation: Georgiev G.Z., "Area of a Square Calculator" , [online] Available at: https://www.gigacalculator.com/calculators/area-of-square-calculator.php URL [Accessed Date: 28 Aug, 2024].

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Area of a square

Here you will learn about finding the area of a square, including counting units squares and multiplying the side lengths.

Students will first learn about the area of a square as part of measurement and data in 3 rd grade.

What is the area of a square?

The area of a square is the amount of space inside the square. It is measured in square units.

For example,

The square units line up with all sides of the square and do not overlap. The area of the square is the number of square units within it.

Notice the connection between the side of a square and the square units:

The side length is always the same as the number of columns and rows of square units, since a square has equal sides.

This is why the area of square formula is \text {side} \times \text {side.}

Area is always labeled in square units. You can write it out or use the exponent 2 (a small 2 ) to show the units are square.

square centimeters (\mathrm{cm}^2), square meters (\mathrm{m}^2), square feet (\mathrm{ft}^2), square inches (\mathrm{in}^2).

The area of other polygons is also measured in square units.

The square units of the rectangle line up with the sides, just like they did for the square. There are 27 square units inside the rectangle, so that is the area.

Notice that this is not true for the other shapes. Some or all of their sides do not line up with the square units. Counting their square units is not as easy. This is why the area of a circle, triangle, hexagon and other shapes is covered in later grades.

Common Core State Standards

How does this relate to 3 rd grade math?

• Grade 3 – Measurement and Data (3.MD.C.5) Recognize area as an attribute of plane figures and understand concepts of area measurement. a. A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area. b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.
• Grade 3 – Measurement and Data (3.MD.C.6) Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).
• Grade 3 – Measurement and Data (3.MD.C.7a) Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.

How to find the area of a square

In order to find the area of a square by counting unit squares:

Be sure that the square is made of whole unit squares in an array .

Count the unit squares to find the area.

In order to calculate the area of a square:

Identify the side length of the square.

Write down the formula for the area of a square.

Substitute the given values and calculate.

Write down your final answer with units squared.

[FREE] Area Check for Understanding Quiz (Grade 4 to 6)

Use this quiz to check your grade 4 to 6 students’ understanding of area. 15+ questions with answers covering a range of 4th, 5th and 6th grade area topics to identify areas of strength and support!

Area of a square examples

Example 1: counting unit squares to find the area.

What is the area of the square?

The unit squares inside the square line up with the sides of the square.

2 Count the unit squares to find the area.

The area of the square is 9 square units.

Example 2: counting unit squares to find the area

The area of the square is 36 square units.

Example 3: counting unit squares (and multiplying the sides) to find the area

Since the length of its sides is 7, the unit squares form a 7 by 7 array.

To find the total number of unit squares (the area) multiply: \text {side} \times \text {side. }

\text { Area }=7 \times 7=49

You can prove this by counting the unit squares within.

The area of the square is 49 square units.

Example 4: counting unit squares (and multiplying the sides) to find the area

Since the length of each side is 5, the unit squares form a 5 by 5 array.

To find the total number of unit squares (the area) multiply: \text {side} \times \text {side.}

\text { Area }=5 \times 5=25

The area of the square is 25 square units.

Example 5: multiplying the sides to calculate the area

Since the length of the sides is the same, all sides are 8 \, cm.

\text { Area of a square }=\text { side } \times \text { side }

\text { Area of a square }=8 \times 8=64

The area of the square is 64 square centimeters.

Example 6: multiplying the sides to calculate the area

The wall above needs to be painted. How many square feet of paint is needed to cover the wall?

Since the length of a side is the same as the others, all sides are 4 \, ft.

\text { Area of a square }=4 \times 4=16

The wall needs 16 square feet of paint to cover it.

Teaching tips for area of a square

• Make sure students have plenty of practice with tiling squares to find the area (via manipulatives or a digital resource that allows them to do this). These opportunities can provide the conceptual understanding necessary to use the area formula.
• Worksheets can be a great source for practice, but be sure to use worksheets that provide a variety of practice (squares whose area can be found by using physical tiles and squares with or without grids).

Easy mistakes to make

• Not using common units When tiling a shape to find the area, it is important that all the tiles are the same size. When solving area problems with real world measurements, it is important that all measurements are in the same unit. Having different size tiles or measurements will make the area calculation incorrect.

Related area of a quadrilateral lessons

• Area of a rectangle
• Area of a parallelogram
• Area of a trapezoid
• Area of a rhombus

Practice area of square questions

1. What is the area of the square?

4 square units

8 square units

12 square units

16 square units

The unit squares inside the square line up with the sides of the square and form an array.

The area of the square is 16 square units.

2. Which square has an area of 25 square units?

The area can also be found by multiplying \text {side} \times \text {side. } Since 5 \times 5=25, a square with the area of 25 square units will have side lengths of 5.

3. What is the area of the square?

6 square units

24 square units

30 square units

36 square units

Since the length of its sides is 6, the unit squares form a 6 by 6 array.

To find the total number of unit squares (the area) multiply: \text { side} \times \text {side.}

\text { Area }=6 \times 6=36

4. What is the area of the square?

77 square units

64 square units

32 square units

Since the length of its sides is 8, the unit squares form an 8 by 8 array.

\text { Area }=8 \times 8=64

The area of the square is 64 square units.

5. What is the area of the square?

81 square feet

9 square feet

18 square feet

36 square feet

\begin{aligned}\text { Area of a square } & =\text { side } \times \text { side } \\\\ & =9 \times 9 \\\\ & =81\end{aligned}

The area of the square is 81 square feet.

6. The area of a square is   4 square inches. What is the side length of the square?

\begin{aligned}\text { Area of a square } &=\text { side } \times \text { side } \\\\ 4 &=\text { side } \times \text { side }\end{aligned}

Think about a number times itself that equals 4…

Since 2 \times 2=4, the square has a side length of 2 inches.

Area of a square FAQs

To calculate the area of a square that has fractional side lengths, you still multiply \text {side} \times \text {side,} so you can either use a calculator or use what you know about multiplying fractions to solve.

Both the area and the perimeter are calculated using the side lengths, but they are not calculated in the same way. The formula for perimeter of a square is 4 \times \text {side length} and the formula for area is \text {side length} \times \text {side length.}

Surface area is the area of the faces of 3D figures. It is found by calculating the area of each side of a shape and then adding them all together. See also : Surface area

The next lessons are

• Surface area
• Pythagorean Theorem

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Python Program to Calculate Area and Perimeter of Square 2 min read

To find the perimeter of a square, just add up all the lengths of the sides.

The area of a square: To find the area of a square, multiply the lengths of two sides together. Another way to say this is to say “square the length of a side.

In this example, I’ll show how to calculate area and perimeter of rectangle in C#.

Python Code:

This code defines a program in Python that calculates the area and perimeter of a square. The program begins by prompting the user to enter the side of the square using the input function and storing the value in a variable s . It then calculates the area of the square by multiplying s by itself and assigns the result to a variable area . It calculates the perimeter by multiplying s by 4 and assigns the result to a variable perimeter . Finally, the program prints the values of area and perimeter to the terminal using the print function.

You may also like: Python Tutorial with Exercises(100+ Examples)

To run this code and see the output, you can follow these steps:

• Save the code to a file or type it into a text editor.
• Open a terminal or command prompt.
• Navigate to the directory where the code is saved.
• Run the command python filename.py , where filename.py is the name of the file containing the code.
• The program will prompt you to enter the side of the square. Type in a value for this variable and press enter.
• The program will then calculate the area and perimeter of the square and print the results to the terminal.

For example, if you enter 5 for the side of the square, the output will be:

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Python Program to Find Area of Square

This article is created to cover some programs in Python, to find and print area of a square based on the length of its side, entered by user at run-time. Here are the list of programs:

• Find Area of Square without Function
• Find Area of Square using User-defined Function
• Using Class and Object

Formula to Find Area of Square

To find area of a square, use:

Here area indicates to area value and len indicates to length value of square

Find Area of Square

To find area of a square in Python, you have to ask from user to enter the side length. Now use the formula to find and print the area value as shown in the program given below:

Here is the initial output produced by this program:

Now supply the input say 6 as side length of square to find and print its area based on the given value of its side length as shown in the snapshot given below:

Find Area of Square using Function

This program is created using a user-defined function named areaOfSquare() . This function takes side length as its argument and returns the area value. The return value of this function gets initialized to a . Now print the value of a as output. That's it:

Here is its sample run with user input, 43.23 as side length of square:

Find Area of Square using Class

This is the last program on finding area of a square, using class and object, an object-oriented feature of Python. The program is little similar to the program created using function. The only difference is, we've to call the member function areaOfSquare() of class using its object through dot (.) operator.

Note - Before accessing any member function of a class using an object say obj , the properties of class must be assigned to it. Therefore through the statement, obj = CodesCracker() , we've done the job.

This program produces the same output as of previous program.

Same Program in Other Languages

• Java Calculate Area of Square
• C Calculate Area of Square
• C++ Calculate Area of Square

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Area of Square

Area of square formula.

The area of a square is the space or region enclosed by the square’s boundary. As you can see in the following illustration, the colored region within the boundary is the area of the square. For emphasis, the borders are colored green, while the area is colored orange/yellow.

Intuitively, the area of a square can be calculated by counting the number of unit squares that are needed to entirely fill or cover the space of the square.

In the diagram below, a large square with a side length of [latex]5[/latex] units can fit [latex]25[/latex] small squares such that each smaller square has a side length of [latex]1[/latex] unit. Loosely speaking, the number of little congruent squares that cover the space within the big square is square’s area. As a result, the area is [latex]25[/latex] square units.

Finding Area of Square Using its Side

How do we find the area of a square using a formula? First, let’s recognize that a square is a special type of rectangle.

Remember, the area of rectangle is the product of its length and width.

[latex]A = lw[/latex]

But in the case of square, the length is equal to the width.

[latex]l = w[/latex]

So, we can express the formula of the area either in terms of the length [latex]l[/latex] or width [latex]w[/latex].

[latex]A = \left( l \right)\left( l \right) = {l^2}[/latex] [latex]\\[/latex]

[latex]A = \left( w \right)\left( w \right) = {w^2}[/latex]

Yet, when discussing squares, the terms “length” and “width” are interchangeable and can lead to misunderstandings because they refer to the same thing. It is preferable to refer to the length or width as simply the side [latex]s[/latex] in order to avoid any confusion.

We can simplify the formula by replacing [latex]l[/latex] for length or [latex]w[/latex] for width by [latex]s[/latex] to mean side.

[latex]l \to s[/latex] [latex]\\[/latex]

[latex]w \to s[/latex]

Therefore, the formula of the area of square becomes

[latex]A = \left( s \right)\left( s \right) = {s^2}[/latex]

Let’s go over some examples!

Examples of Finding Area of Square on a Grid

Example 1: Find the area of the square below on a grid.

To find the area of a square on a grid paper, simply count the number of smaller squares contained within the larger square whose area you wanted to determine.

Let’s begin at 1 and keep going until we’ve counted all the squares. We will record the numbers in the squares to keep track of the total number of squares.

Since there are 36 small squares that cover the big square, it implies that the area of the square is 36 square units .

Example 2: What is the total area of the two squares on the grid?

This is a very similar problem to Example 1. The only difference is that we now have two squares instead of one. Since we are interested in finding the total areas of the two squares, we must first determine the area of each square and then add them together to arrive at the final answer.

Let’s count the number of little squares in each huge square, the green and blue squares. Please refer to the diagram below.

Because the green square comprises 25 little squares, its area is 25 square units . The blue square, on the other hand, has 9 little squares, thus its area is 9 square units . Finally, to get the overall area, add the areas of the green and blue squares which gives us 25 + 9 = 34. Therefore, the total area is 34 square units .

Examples of Finding the Area of Square using its Side

To find the area of a square when the side [latex]s[/latex] is given, simply multiply the side by itself [latex]s \times s[/latex] or square it [latex]s^2[/latex].

Example 3: Find the area of a square whose side length is [latex]7[/latex] [latex]ft[/latex].

A square is a specific case of a rectangle in which the length [latex]l[/latex] and width [latex]w[/latex] are the same. Since the length and width have the same measure, let’s simply call it as side [latex]s[/latex]. Below is the illustration of the given problem. Notice, both sides have the same measure of [latex]7[/latex] [latex]ft[/latex].

To find the area of the square, we multiply the two sides together.

[latex]A = s \times s = 7 \times 7 = 49[/latex][latex]\,[/latex]unit[latex]^2[/latex]

Or simply square the side. It will have the same result.

[latex]A = {s^2} = {\left( 7 \right)^2} = 49[/latex][latex]\,[/latex]unit[latex]^2[/latex]

Therefore, the area of the square is 49 square units.

Example 4: The side of a square is [latex]1[/latex] yard. What is the area of the square in square inches?

Don’t jump to the conclusion that the solution is [latex]1[/latex] square yard since side times side equals 1, that is, [latex]\left( 1 \right)\left( 1 \right) = 1[/latex]. If the problem is asking for the area in square yards, you are correct. But this is not the case. Clearly, the question is asking for the area in terms of square inches not in square yards.

That is why we must approach each problem with caution. The key is to thoroughly and carefully read the question and to ensure that we understand what is being asked in order to answer the problem correctly!

The first thing we should do is to convert the given unit of measure into the desired unit which is from yard to inch. That means we will have to convert [latex]1[/latex] yard into inches. We know that [latex]1[/latex] yard equals [latex]3[/latex] feet but [latex]1[/latex] foot equals [latex]12[/latex] inches.

Now, let’s convert 1 yard into inches:

[latex]1\,yd\,\, \times \,\,{\Large{{{3\,\,ft} \over {1\,yd}}}}\,\, \times \,\,{\Large{{{12\,\,in.} \over {1\,ft}}}}\,\, = \boxed{\,\,36\,\,in.}[/latex]

Therefore, [latex]1[/latex] yard is equal to [latex]36[/latex] inches.

Upon knowing the side length of the square in terms of inches, we are ready to compute for the area of the square using the formula.

Since the side is [latex]36[/latex] inches, we have

[latex]A = {s^2} = {\left( {36} \right)^2} = 1,296[/latex] [latex]\,[/latex][latex]in^2[/latex]

Therefore, the area of the square is [latex]1,296[/latex] square inches or [latex]\text{in}^2[/latex]

Examples of Finding the Area of Square using its Diagonal

In previous examples, we calculate the area of square using its side [latex]s[/latex]. Is there any other way to find its area other than its side? Well, there is! The fact is, it is possible to solve for the area of square if we know its diagonal. But before I simply give the formula, it is much better if I derive it for you so you can get a better appreciation why it works.

Starting off, let’s derive the formula for calculating a square’s area given its diagonal [latex]d[/latex]. Here we have a square where the sides are labeled as [latex]s[/latex] and the diagonal is labeled as [latex]d[/latex]. Notice that the two sides of the square and its diagonal form a right triangle (a triangle that contains a 90-degree angle).

It implies we can mathematically relate the diagonal and the sides of the right triangle in an equation using the Pythagorean Theorem .

The square of the length of the hypotenuse [latex]c[/latex], which is the side opposite the right angle, is equal to the sum of the squares of the lengths of the other two sides which are the legs namely [latex]a[/latex] and [latex]b[/latex].

In our right triangle, the legs have the same side length which is designated by [latex]s[/latex]. In addition, the hypotenuse (the side opposite the 90-degree angle or the longest side) is represented by the letter [latex]d[/latex] to mean diagonal.

Substitute both [latex]a[/latex] and [latex]b[/latex] by [latex]s[/latex], and [latex]c[/latex] by [latex]d[/latex].

Notice below that we combined [latex]s^2[/latex] and [latex]s^2[/latex] to get [latex]2s^2[/latex].

To solve for [latex]s^2[/latex], we divide both sides of the equation by [latex]2[/latex].

Notice, I didn’t completely solve for [latex]s[/latex] by taking the square root of both sides. I intentionally kept [latex]s^2[/latex] because that is actually what we want since the area of the square is side times side or simply side squared, [latex]s^2[/latex].

Therefore, an alternative formula to find the area of the square is

[latex]A = {\Large{{{d^2}} \over 2}}[/latex]

where [latex]d[/latex] is the diagonal of the square

Example 5: What is the area of a square whose diagonal is [latex]8[/latex] [latex]\text{ft}[/latex]?

The good thing is that we have already done the hard work above! We have derived the formula to find the area of a square when the diagonal of the square is given to us.

Since the diagonal is [latex]8[/latex] [latex]\text{ft}[/latex] as you can see in the illustration below, we should be able to find the area of the square by simply substituting the value of the diagonal into the formula and then simplify.

The square of [latex]8[/latex] is [latex]64[/latex]. Then we divide it by [latex]2[/latex] to get [latex]32[/latex].

Therefore, the area of the square is [latex]32[/latex] [latex]\text{ft}^2[/latex].

The formula above can also be used to calculate the diagonal of a square if we know its area. Let’s look at an example.

Example 6: If the area of a square is [latex]72[/latex] [latex]\text{cm}^2[/latex], what is the length of the diagonal of the square?

Let’s begin by the writing the formula of the area of the square when the diagonal is given.

[latex]\boxed{A = {{{d^2}} \over 2}}[/latex]

Since we know the area of the square, let’s replace [latex]A[/latex] by [latex]72[/latex] on the left side of the equation.

To keep [latex]d^2[/latex] to the right side of the equation, we will multiply [latex]2[/latex] on both sides of the equation.

Finally, to isolate the variable [latex]d[/latex] on the right side we will take the square roots of both sides.

Therefore, the diagonal of the square is [latex]12[/latex] [latex]\text{cm}[/latex].

Examples of Finding the Area of Square using its Perimeter

Now there is a case when we are given the perimeter of the square and we are asked to find its area. How do we do that?

Let’s begin by recalling that the square’s perimeter is merely the sum of all its sides. Since a square is a quadrilateral, it has four (4) sides, so to calculate its perimeter, we must add all four sides together. Moreover, the sides of a square are of equal length. If we let one side to be [latex]s[/latex] then the sum of all four sides must be [latex]s+s+s+s=4s[/latex].

Therefore, the formula for the perimeter of a square is

[latex]\boxed{P = s + s + s + s = 4s}[/latex]

Having learned how to find the square’s perimeter if we know its side, it should be possible to use the same formula to determine the side of the square if the perimeter is given to us.

Looking at the formula again where the perimeter of square is four times its side, [latex]4s[/latex]. The variable [latex]s[/latex] [which is the side of the square] can be solved or isolated by dividing both sides of equation by [latex]4[/latex].

[latex]\boxed{P = 4s}[/latex]

If we divide the entire equation by [latex]4[/latex], we get:

Simply put, dividing the perimeter by [latex]4[/latex] will yield the side length of the square. That’s it!

Once we know the measure of one side of the square (note: we only need one since all sides are congruent), we can go back to the basic method for calculating the area of a square, which is side times side [latex]\text{side}[/latex][latex]\times[/latex][latex]\text{side}[/latex] or side squared, [latex]\text{side}^2[/latex].

[latex]\text{Area of Square}\,\square\,[/latex][latex]=[/latex][latex]\,\text{side}[/latex][latex]\times[/latex][latex]\text{side}\,[/latex][latex]=[/latex][latex]\,s^2[/latex]

The best way to illustrate this procedure is through an example!

Example 7: Find the area of a square whose perimeter is [latex]56[/latex] meters.

Let’s draw a rough sketch of a square with a perimeter of [latex]56[/latex] [latex]\text{m}[/latex].

Since a perimeter is simply the sum of all sides, and a square has four (4) equal sides, the formula of the perimeter of square must be

[latex]P = s + s + s + s = 4s[/latex]

What we want is to calculate the side of the square given its perimeter. That can be achieved by simply dividing the perimeter by [latex]4[/latex].

Upon knowing from our calculation above that the side of the square is [latex]14[/latex] meters, we can now find its area by substituting the measure of the side to the fundamental formula of the area of square which is the square of its side , [latex]s^2[/latex].

Therefore, the area of the square is [latex]196[/latex] [latex]\text{m}^2[/latex].

Example 8: What is the area of a square with perimeter [latex]12[/latex] feet?

We should be able to solve this problem very easily with the technique above by simply dividing the given perimeter by [latex]4[/latex] and squaring the quotient to get the area of the square.

But let’s derive a much simpler formula that we can add in our math toolbox.

Remember, if we know the perimeter of the square, we can divide by [latex]4[/latex] to get the measure of the side of the square.

That means, if we square both sides of the equation, the right side becomes [latex]s^2[/latex] which is literally the area of the square. In other words, we can also determine the area of square by squaring the perimeter and dividing it by [latex]16[/latex].

Thus, a much simpler formula to find the area of a square when its perimeter is known:

[latex]\boxed{A = {{{P^2}} \over {16}}}[/latex]

So, if a square has perimeter of [latex]12[/latex] feet, its area must be

Therefore, the area of the square is [latex]9[/latex] [latex]\text{ft}^2[/latex].

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How to Find the Area of a Square

Last Updated: July 24, 2023 Fact Checked

This article was co-authored by David Jia . David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 286,403 times.

Finding the area of a square is easy as long as you know the length of its side, perimeter, or diagonal. Here's how to do it.

Using the Length of a Side

• Squaring the side of a square is the same thing as multiplying the square's height times its base.

Using a Known Diagonal

Using a Known Perimeter

Community Q&A

You Might Also Like

• ↑ https://www.mathplanet.com/education/pre-algebra/inequalities-and-one-step-equations/calculating-the-area-and-the-perimeter
• ↑ https://www.youtube.com/watch?v=OC2rMWzc-I8
• ↑ https://sciencing.com/area-square-using-its-perimeter-7991359.html

About This Article

To find the area of a square, use the formula a = side^2, where side is the length of one of the sides of the square. If you only know the perimeter of the square, you can find the area by dividing the perimeter by 4, which will give you the length of each side, and then plugging the side into the formula a = side^2. If you want to learn how to find the area of a square if you only know the length of a diagonal, keep reading! Did this summary help you? Yes No

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