Area of any figure is defined as the space occupied by it in 2-D space. Similarly, the Area of the square is defined as the space enclosed by the boundary of the square. The measurement of the area is done in square units. The SI unit for measurement of the area is m2. For finding areas of various figures, several pre-defined formulas are used, in this article, we will study the formulas for finding the area of the square.

## What is Area?

Space enclosed inside the boundaries of any figure is called the area of the figure. It is a physical quantity that gives us the idea of how much space is covered by an object. Square is a Two-Dimensional (2D) figure which has 4 sides of all equal length. The area of a square concept comes under the topic of mensuration which deals with the measurements of Two Dimensional and Three Dimensional figures. Length, Perimeter, Area, Volume, etc. come under measurements of a figure.

## Area of Square

The area is the region inside the boundaries of an object. The area of a square is defined as the number of square units needed to fill the square. To calculate the square area we need to know the length of any of its sides. The area of the square can be calculated by squaring the length of any of its sides.

## How to Find Area of a Square?

Area of a square can be calculated if the dimensions of the square are known. We can calculate the area of the square by various formulas depending on the initial values given. The various formulas for finding the area of a square are listed below:

### Area of Square (if Sides are given)

Area of Square = Side2 unit2

Area can be measured in various units, some of the conversions for changing standard units of the area to other desired units are given below: 1 m2 = 10000 cm2 1 km2 = 1,000,000 m2 If required in calculation we can find the perimeter of the square by the given formula

Perimeter of square = 4 × sides units

Example: What is the area of a square if each side of length is 4cm? Solution:

Given Side length (s) = 4cm Area of square = s2 = 42 = 16cm2 Area of square with side length 4cm is 16cm2.

### Area of Square (if Diagonal is Given)

Area of Square when the diagonal length is given,

Area = (1/2) × d2 where, d is the length of diagonal.

Example: Find the area of a square if the length of the diagonal is 6cm. Solution:

Given Diagonal length (d) = 6cm Area = (1/2) × d2 = (1/2) × 62 = 36/2 = 18cm2 Area of square is 18cm2.

### Area of Square (if Perimeter is Given)

We can find an area even if the perimeter of a square is given. The formula of the perimeter of a square = 4 × side From the above formula, we can find the side length by dividing Perimeter by 4. Side length(s) = Perimeter/4 Using side length we can find the area of the square by using the formula Area = side × side. Example: Find the Area of the Square if the perimeter of a square is 36 cm. Solution:

Given Perimeter=36 cm So, Side length=perimeter/4 Side(s) = 36/4 = 9 cm From the side length we can calculate area of square by Area=Side2 = 92 = 81 cm2 Area of square with perimeter 36 cm is 81 cm2.

## Solved Examples on Area of Square

Example 1: Find the Area of the Square if the perimeter of a square is 64cm. Solution:

Given Perimeter=64cm So, Side length=perimeter/4 Side(s) = 64/4 = 16cm From the side length we can calculate area of square by Area=Side2 = 162 =256cm2 Area of square with perimeter 64cm is 256cm2.

Example 2: Find the area of a square if the length of the diagonal is 12cm. Solution:

Given Diagonal length (d) = 12 cm Area = (1/2) × d2 = (1/2) × 122 = 144/2 = 72 cm2 Area of square is 72 cm2.

Example 3: The length of each side of a square is 5cm and the cost of painting it is Rs. 5 per sq. cm. Find the total cost to paint the square. Solution:

Given Side length (s) = 5cm Area of square = s2 = 52 = 25cm2 For 1 sq.cm cost of painting is Rs 5. Total Cost of painting the 25sq cm= 25 × 5 = Rs125

Example 4: A floor which is 60 m long and 30 m wide is to be covered by square tiles of side 6 m. Find the number of tiles required to cover the floor. Solution:

Length of the floor = 60 m Breadth of the floor = 30 m Area of floor = length × breadth = 60 m × 30 m = 1800 sq. m Length of one tile = 6 m Area of one tile = side ×side = 6 m × 6 m = 36 sq. m No. of tiles required = (area of floor)/(area of one tile) = 1800/36 = 50 tiles. Total tiles required is 50.

Example 5: What is the Area of a Square if the perimeter of a square is 24 cm? Solution:

Given, Perimeter = 24cm So, Side length = perimeter/4 Side(s) = 24/4 = 6cm From the side length we can calculate area of square by Area = Side2 = 62 = 36cm2 Area of square with perimeter 24 is 36cm2.

## FAQs on Area of Square

Question 1: What is the Area of the square? Answer:

Area of a square is defined as the total number of units of a square that is enclosed by the boundary of a square. i.e. it is defined as the space occupied by the square in the 2-D plane.

Question 2: What is the Formula for the area of a square? Answer:

A square is a quadrilateral with all four sides equal. Its area can be calculated with formula Sides square, i.e. Area of Square is side × side square units.

Question 3: What is the standard unit for measuring the area of a square? Answer:

Area of square is measured in square units i.e. square m, square cm, etc.

Question 4: What is the formula for finding the area of a square if a diagonal is given? Answer:

Suppose the diagonal of a square is given, then the formula to find the area of a square is given by: Area = (½) × d2 square units where, “d” is the diagonal

### Related Articles

• Area of Circle
• Area of Rectangle
• Area of Trapezium

How do you calculate the area of a square if the perimeter is given? The perimeter of the square is the sum of all four sides of the square. If the perimeter is given, then the formula to calculate the area of the square, A = Perimeter2/16

## Practice Problems

For example, So, the area occupied by 200 tiles of sides 30 inches = Area of one tile x Total number of tiles Given: Example 1: Given that each side is 5 cm, find the area of a square. = 18 Total Floor Area = 900 x 200 square inches Here, the length of the diagonal = 6 ft Correct answer is: 18 ft
The formula to calculate the area of the square when the diagonal is given is d2/2. The area of a square when the diagonal, d, is given is d2÷2 square units. Substituting the diagonal value, we get:

## Conclusion

For example, The space occupied by the swimming pool below can be found by finding the area of the pool. = 900 square inches Thus the cost of painting a 2500 sq. m wall = Rs. 2 × 2,500 = Rs 5,000 Example 2: The side of a square wall is 50 m. What is the cost of painting it at the rate of Rs. 2 per sq. m? Area of a square = side × side What are the units of the area of the square? The area of the square is 2-dimensional. Thus, the area of the square is always represented by square units, for which the common units are cm2, m2, in2, or ft2. We know that the formula to find the area of a square when the diagonal, d, is given is d2÷2 square units. Area = 5 × 5 As per the formula, it can be written as: The area of a square is equal to (side) × (side) square units. Do two squares of equal areas have equal perimeters? Yes. Two squares of equal areas, given by side x side, will have the same side lengths. They are congruent. Consequently, the perimeters of the two squares, given by 4 x side length, will be equal as well. Thus, the area of the square is 8 cm2. = 42÷2 = 16 ÷ 2 = 8 Area = 25 cm2 Area of 1 tile = 30 x 30 inches
= 180000 sq inches 24 meters 26 meters 28 meters 30 meters Correct answer is: 28 meters
Answer: Area of garden = 784 m
Side of the garden = ?
We know the formula to calculate the side of the square or garden, A = Side
It can be written as,
784 m = Side
Therefore, side of the garden =
= 28 meters Correct answer is: 324 m
324 m

Side of the swimming pool, or a = 18m

= 18 x 18

## Solved Examples

Area of swimming pool = a
Side of one tile = 30 inches The area of a square with each side 8 feet long is 8 × 8 or 64 square feet (ft2). In the given square, the space shaded in violet is the area of the square. Side of the wall = 50 m The cost of painting 1 sq. m = Rs. 2 To learn similar concepts, head over to SplashLearn. The game-based learning platform has interactive games, worksheets, and courses that make learning fun and intriguing. Enjoy the flexible time schedule, and get the best-curated knowledge from highly qualified professional teachers. Side, d = 4 cm Thus, the area of the square table is 18 sq feet. 180000 sq inches 180000 sq meters 220000 sq inches 240000 cubic inches Correct answer is: 180000 sq inches
Total number of tiles = 250 The number of square units needed to fill a square is its area. In common terms, the area is the inner region of a flat surface (2-D figure). = 36/2
Area of Square= 6 /2
= 324 m Solution: Area of the wall = side × side = 50 m × 50 m = 2500 sq. m Example 3: Find the area of a square whose diagonal is measured is 4 cm. What is the difference between the perimeter and area of a square? The perimeter of a square is the sum of its four sides or the length of its boundary. It is a one-dimensional measurement and expressed in linear units. Area of a square is the space filled by the square in two-dimensional space. It is expressed in square units. Area of a square is defined as the number of square units needed to fill a square. In general, the area is defined as the region occupied inside the boundary of a flat object or 2d figure. The measurement is done in square units, with the standard unit being square metres (m2). For the computation of area, there are pre-defined formulas for squares, rectangles, circles, triangles, etc. In this article, you will learn about the area of a square. The area is the space covered by the object. It is the region occupied by any shape. While measuring the area of a square, we consider only the length of its side. All sides of a square are equal; hence, its area is equal to the square of the side. Similarly, we can find the area of the other shapes such as rectangles, parallelograms, triangles or any polygon, based on its sides. The area of the surface is calculated based on the radius or the distance of its outer line from the axis for curved surface objects.

## Area of a Square Formula

Before moving into the area of square formula used for calculating the region occupied, let us try using graph paper. You are required to find the area of a side 5 cm. Using this dimension, draw a square on a graph paper having 1 cm × 1 cm squares. The square covers 25 complete squares. Thus, the area of the square is 25 square cm, which can be written as 5 cm × 5 cm, that is, side × side. From the above discussion, it can be inferred that the formula can give the area of a square is: Area of a Square = Side × Side Therefore, the area of square = Side2 square units and the perimeter of a square = 4 × side units Here some of the unit conversion lists are provided for reference. Some conversions of units:

• 1 m = 100 cm
• 1 sq. m = 10,000 sq. cm
• 1 km = 1000 m
• 1 sq. km = 1,000,000 sq. m

## Area of a Square Sample Problems

Example 1: Find the area of a square clipboard whose side measures 120 cm. Solution: Side of the clipboard = 120 cm = 1.2 m Area of the clipboard = side × side = 120 cm × 120 cm = 14400 sq. cm = 1.44 sq. m Example 2: The side of a square wall is 75 m. What is the cost of painting it at the rate of Rs. 3 per sq. m? Solution: Side of the wall = 75 m Area of the wall = side × side = 75 m × 75 m = 5,625 sq. m For 1 sq. m, the cost of painting = Rs. 3 Thus, for 5,625 sq. m, the cost of painting = Rs. 3 × 5,625 = Rs 16,875 Example 3: A courtyard’s floor which is 50 m long and 40 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 = 50 m The breadth of the floor = 40 m Area of the floor = length × breadth = 50 m × 40 m = 2000 sq. m Side of one tile = 2 m Area of one tile = side ×side = 2 m × 2 m = 4 sq. m No. of tiles required = area of floor/area of a tile = 2000/4 = 500 tiles.

## Practice Problems

1. A square wall of length 25 metres, has to be painted. If the cost of painting per square metre is ₹ 4.50. Find the cost of painting the whole wall.
2. Find the length of a square park whose area is 3600 square metres.
3. Find the area of the square whose length of the diagonal is 5√2 cm.

To learn and practice more problems related to the area of a square, download BYJU’S-The Learning App.

## Frequently Asked Questions on Area of Square

### What is the area of a square?

As we know, a square is a two-dimensional figure with four sides. It is also known as a quadrilateral. The area of a square is defined as the total number of unit squares in the shape of a square. In other words, it is defined as the space occupied by the square.

### Why is the area of a square a side square?

A square is a 2D figure in which all the sides are of equal measure. Since all the sides are equal, the area would be length times width, which is equal to side × side. Hence, the area of a square is side square.

### What is the area of a square formula?

The area of a square can be calculated using the formula side × side square units.

### How to find the area of a square if a diagonal is given?

If the diagonal of a square is given, then the formula to calculate the area of a square is:
A = (½) × d2 square units.
Where “d” is the diagonal

### What is the perimeter and the area of a square?

The perimeter of the square is the sum of all the four sides of a square, whereas the area of a square is defined as the region or the space occupied by a square in the two-dimensional space.

### What is the area of a square if its side length is 10 cm?

Given: Side = 10 cm
We know that, Area of a square = Side × Side square units
Thus, Area = 10 × 10 = 100 cm2
Therefore, the area of a square is 100 cm2 if its side length is 10 cm.

### What is the unit for an area of square?

The area of a square is measured in square units.

### How to calculate the area of a square if its perimeter is given?

Follow the below steps to find the area of a square if its perimeter is given:
Step 1: Find the side length of a square using the perimeter formula, P = 4 × Side
Step 2: Substitute the side length in the area formula: A = Side × Side.

### 3. How Do You Calculate the Area of a Square?

Solution: We are given, diagonal = 13 cm Ans. Area of the square formula is a way to calculate the area of a square. The formula for the area of a square is A = s2, where s represents the length of each side of the square.

• Step 1: Note the perimeter of the given square.
• Step 2: We know the value of the perimeter of a square is 4s. Therefore 4s = Perimeter.
• Step 3: Substitute the value of perimeter and find the side using the formula s = Perimeter/4
• Step 4: Now that we know the side of the square. Find the area using s2.
• Step 5: Write the answer in square units.

Area of the square = (4)2/2 = 16/2 = 8 cm2 In the above sections, we learned how to calculate the area of a square when either the side or diagonal is given. But, suppose you are not provided with any of these parameters, but the perimeter of the square is given. How will you find the area when the perimeter of square is given? Let us find out: From the above illustration, we learn that the area of a square is equal to the product of the sides. This can be written as ‘side x side’. Hence the formula for any square with any length of a side is given as

• All four sides are the same.
• All angles are 90o.

Take note of the following factors to keep in mind when you calculate the area of a square. Solution: Given that length of the table = 8 m Let us look at an example to understand this formula: = 169/2 cm2

### 4. What are the Units of the Area of a Square?

Example: Find the area of a square with a diagonal length of 4 cm Squares can be found everywhere. Here are some examples of commonly seen square-shaped objects. A square is represented by the chessboard, the clock, a blackboard, and a tile.

### 5. What is the Perimeter and Area of Square Formulas?

The formula for the area of a square is: A = s2 Don’t worry! The area of the square can be evaluated even if the length of the diagonal is given. You can find the area using the formula written below: Side (s) = perimeter/4 = 162

### Area of a Square Formula

The area of square = (Side)2 = (13)2 = 169 cm2. Example: Find the area of a square when the diagonal length is 13 cm.

• Step 1: Note down the value of the side, say ‘a’.
• Step 2: Substitute the value of a in the formula -> Area (with side) = (Side)2 = (a)2
• Step 3: Write the answer in square units.

Area of square ( using diagonals) = (D)2/2, where D represents the diagonal length. First, we will figure out the length of the sides of the garden. While evaluating the area of a square, we often mistake doubling the number. This is not the case! Keep in mind that the area of a square is not 2 x side. It is always either ‘side x side’ or side2.

#### Find Area of Square When the Perimeter of a Square is Given

= 16 cm = (13)2/2 Solution: Given that length of the diagonal = 4 cm Ans. The units for the area of a square are squared units. The most common unit for measuring area is square metres, but sometimes you might see square feet or acres as well. What if we are not provided with the sides of a square but rather we are given the diagonal length? How do we find the area of a square in this case? Other dimensions, such as the diagonal and the perimeter of the square, can also be used to compute the area of a square. In this article, we’ll try to learn more about the area of the square. Example: A square garden has a perimeter of 64 cm. Max wants to plant flowers and find the area of this garden but doesn’t know how to do it? Help him figure out the area of the garden. = 256 cm2 The area of a square can be understood by how much space a square covers inside it. In simple terms, the space present within the boundary of a square is known as the area of the square. In this article, you shall learn the fundamental parameters of a square. Also, you will study how to find the area of a square, the area of the square formula, and the surface area of a square pyramid. In our day-to-day life, we can find squares everywhere. From our homes to our schools, squares are present at each corner. The tiles in your kitchen are square. The chessboard is a square containing 64 black and white smaller squares. The most common example is a Rubrik’s cube. Each surface of a Rubik’s cube is square. Note: Remember that the square’s diagonals are equal, so the area remains the same if any of the diagonals are given.

### 2. What is the Area of a Square Formula?

A square is a two-dimensional closed shape that has four equal sides and four equal angles. The four angles at the vertices are formed by the square’s four sides. The perimeter of a square is the sum of the total lengths of its sides, and the area of the square is the total space occupied by the shape. It has the following properties as a quadrilateral. Illustration: Let us consider a square of length 4 units. Now consider smaller squares of length 1 unit each. As we can see in the figure below, 4 squares of 1 unit fill the first row of the larger square. Similarly, 4 squares of 1 unit fill the second, third and fourth row. Now the larger square is filled. If we count the number of smaller squares, we get that 16 squares of 1 unit fill the square of 4 units. Hence 16 units are the area of the square. Therefore the area of plastic required to cover the table = area of the table. Area of the garden = (s)2

1. When anyone diagonal is given:
1. When anyone’s side is given:

Example: Find the area of plastic required to cover a square table of length 8 m. Solution: We know: Perimeter of the garden = 64 cm

• A square has all sides equal. This implies that the opposite and the adjacent sides of a square are equal to each other.
• The opposite sides of a square are parallel, making it a parallelogram.
• The adjacent sides of a square are perpendicular to each other. This means that any two adjacent sides have an angle of 90 degrees between them.
• A square is divided into two right-angled congruent triangles.
• A square is a special case of a rectangle.
• The perimeter of a square: The distance covered by the boundaries of a square is known as the perimeter of a square. It is formulated as:

The area of a square is always in square units ( square cm, square m, square inches, etc.)

## Area of Square

#### Some tips from our side:

Example: Find the area of a square with sides of a length of 13 cm. Are we all familiar with what a square is? A square is a closed quadrilateral. Quadrilaterals are figures having 4 sides. Thus square is a four-sided figure which has all four sides equal. If one side of a square is 10 cm, then the other sides are also equal to 10 cm. Let us learn some of the mathematical terms and concepts related to a square first: The two sides are parallel. Solution: Given the length of the side = 13 cm

## What is the Area of the Square?

We must remember to write the area’s unit when representing it. The area of a square is always two-dimensional; hence we use square units. For example cm2, m2, inch2, etc. Area = (Side)2 = 64/4 Hitherto, we have learned 2 formulas related to finding the area of a square. Let us learn how you will approach the questions related to the area of any square. Using the formula in step 3 Area of table = (side)2 = 82 = 64 m2 The area of a square is defined as the number of square units required to fill this shape. In other words, when calculating the area of a square, we consider the length of its side. Because all of the sides of the shape are equal, its area is the product of its two sides. The most common units for measuring the area of a square are square meters, square feet, square inch, and square cm. Ans. The formula for the perimeter of a square is: p = 4s The area of a square can also be calculated using other dimensions, such as the diagonal and the perimeter of the square. On this page, we’ll try to learn more about the area of a square. Ans. Area of the square in geometry is the measurement of a surface. It is calculated by multiplying the length by the width. Perimeter (square) = s + s + s + s = 4 x s = 4s {where ‘s’ represents the side of a square}

• Step 1: Write down the value of the diagonal length, say ‘d’.
• Step 2: Substitute the value of d in the formula -> Area (with diagonal) = (d)2/2 =
• Step 3: Write the result in square units.

Ans. To calculate the area of a square, you need to multiply the length of one side by itself. So if you have a square with sides that are each 10 centimetres long, you’d do: 10 * 10 = 100. Now, Area of the square = D2/2

### 1. What is Area of Square in Geometry?

From this, we can deduce that the area of a square is equal to the product of its two sides. As we know, 16 = 4 x 4, and 4 units form one side of the square. In the next section, we will learn and derive the area of a square formula.

Area of any geometrical figure is the space occupied by a two-dimensional object. A square is a two-dimensional geometric shape that is determined by the sides which are all equal in length and perpendicular to each other (angle between two sides is 90 degrees). The area of a square is the number of square units required to fill a square fully. There are different ways to calculate the area of a square.

One of the conventional and standard ways to calculate the area of the square is by using its diagonals or by using its sides. Since all the sides of a square are the same, we can directly find the square of its side. Therefore, the area of a square is equal to the product of any of its two sides.

But sometimes the length of the side is not given and all we know is the length of the square’s diagonal. With the knowledge of right triangles, we can find the area of a square using diagonal.

### What is the Diagonal of a Square?

A diagonal is a line that stretches from one corner of a figure to the opposite corner, passing through the center of the figure. The diagonals of a square are always equal to each other. In a polygon, the diagonals can be defined as a line joining its two nonadjacent vertices.

### The Relation Between Diagonal and Side of Square

A square can be divided into two right triangles, where the diagonal of the square is equal to the length of the hypotenuse of the triangle. Pythagoras theorem, which applies to right-angled triangles, shows the relation between the hypotenuse and sides of a right triangle.

Thus, it also represents the relation between the diagonal of a square (the hypotenuse of the triangle) and its sides.

Using Pythagoras Theorem,

(Hypotenuse)2= (Base)2+ (Perpendicular)2

Here, the length of the base is equal to the length of perpendicular which is denoted by ‘a’ and hypotenuse is equal to the diagonal which is denoted by ‘d’.

Therefore, a2+ a2 = d2

Diagonal = $\sqrt{a^{2} + a^{2}}$

= $\sqrt{2a^{2}}$

= $a\sqrt{2}$

= side $\sqrt{2}$

### Formula of the Area of Square Using Diagonal

Using the length of the diagonal, the area of a square can be calculated as:

Area of square = ½ × d2 units2

Here, “d” is the length of any of the diagonals. Also, remember that in a square, diagonals are equal.

### Derivation of the Area of Square Using Diagonal

We know the formula to find the area of a square using diagonals. Now, we will derive that formula using the following two methods.

1. Using Pythagoras Theorem

In the given figure, the diagonal of length “d” units divide the square of the side “a” units into two right triangles. Now, applying Pythagoras theorem in any right-angled triangle,

(Hypotenuse)2= (Base)2+ (Perpendicular)2

Here,

Perpendicular = a

Base = a

Hypotenuse = d

So,

a 2 + a 2 = d 2

2a 2 = d 2

Or, a2 = d2/2

We know that area of a square = a2 = d2/2

Thus, area of a square using diagonals = ½ × d2 square units.

1. Using Relation between Side and Diagonal

For a square of side length ‘a’ and diagonal length ‘d’, we know,

Area of a square = side x side = a2

Now, as we have derived above,

Diagonal of square = side x √2 = a√2

Then, side of square, a = 1/√2 x diagonal = d/√2

Thus, area of square = a2

Area = (d/√2)2

Area = d2/2

Area = ½ x d2

Area = ½ x (diagonal)2

Thus, area of a square using diagonals = ½ × d2 square units.

### Solved Examples

Example 1: Find the sides and area of a square when diagonal is given as 6 cm.

Solution: Let us take a square of side x. If the square is divided into two right-angled triangles then the hypotenuse of each triangle is equal to the diagonal of the square. As given, the diagonal is equal to 6 cm.

According to Pythagoras theorem,

x 2 + x 2 = 6 2

2x2 = 36

x 2 = 18

x =$\sqrt{18}$

x = 3$\sqrt{2}$ units

Thus, the length of the side of a square is 3$\sqrt{2}$ units.

To find the area of a square, when diagonal is given, we can use any of the below methods:

1. Method 1

Area of a square = side x side =3$\sqrt{2}$ x 3$\sqrt{2}$

= 9 x 2 = 18 cm2

1. Method 2

Area of a square = ½ x d2 = ½ x 6 x 6

= ½ x 36 = 18 cm2

Example 2: Find the length of the diagonal of a square using the Pythagoras theorem if the sides are 4 cm.

Solution: We know that all the sides of a square are equal in length. We also know that each vertex makes an angle of 90°. Now, let’s split the square into two right triangles, with sides equal to 4 cm. Using the Pythagoras theorem in one of the triangles, we will find the third side of the triangle, which is the diagonal of the square.

Let the hypotenuse/ diagonal be ‘c’ cm.

Therefore, (Hypotenuse)2= (Base)2+ (Perpendicular)2

c2= 42 + 42

= c2= 16 + 16

c = $\sqrt{32}$ cm

c = 4$\sqrt{2}$cm

The length of the diagonal is 4$\sqrt{2}$cm