Area of a Square – Definition with Examples

Definition

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).

In the given square, the space shaded in violet is the area of the square.

For example, The space occupied by the swimming pool below can be found by finding the area of the pool.

The Formula for the Area of A Square

The area of a square is equal to (side) ×  (side) square units.

The area of a square when the diagonal, d, is given is d²÷2 square units.

For example,

The area of a square with each side 8 feet long is 8 × 8 or 64 square feet (ft²).

Solved Examples

Example 1: Given that each side is 5 cm, find the area of a square.

Solution: 

Area of a square = side × side

Area = 5 × 5

Area = 25 cm²

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?

Solution:

Side of the wall = 50 m

Area of the wall = side × side = 50 m × 50 m = 2500 sq. m

The cost of painting 1 sq. m = Rs. 2

Thus the cost of painting a 2500 sq. m wall = Rs. 2 × 2,500 = Rs 5,000

Example 3: Find the area of a square whose diagonal is measured is 4 cm.

Solution:

Given:

Side, d = 4 cm

We know that the formula to find the area of a square when the diagonal, d, is given is d²÷2 square units.

Substituting the diagonal value, we get:

= 4²÷2 = 16 ÷ 2 = 8

Thus, the area of the square is 8 cm².

Frequently Asked Questions

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.

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

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.

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.

Square Formula

Square is a regular quadrilateral. All the four sides and angles of a square are equal. The four angles are 90 degrees each, that is, right angles. A square may also be considered as a special case of rectangle wherein the two adjacent sides are of equal length.

In this section, we will learn about the square formulas – a list of the formula related to squares which will help you compute its area, perimeter, and length of its diagonals. They are enlisted below:

Where ‘a’ is the length of a side of the square.

Properties of a Square

• The lengths of all its four sides are equal.
• The measurements of all its four angles are equal.
• The two diagonals bisect each other at right angles, that is, 90°.
• The opposite sides of a square are both parallel and equal in length.
• The lengths of diagonals of a square are equal.

Derivations:

Consider a square with the lengths of its side and diagonal are a and d units respectively.

Formula for area of a square: Area of a square can be defined as the region which is enclosed within its boundary. As we mentioned, a square is nothing a rectangle with its two adjacent sides being equal in length. Hence, we express area as:

The area of a rectangle = Length × Breadth

Here,

The formula for the perimeter of a square: Perimeter of the square is the length of its boundary. The sum of the length of all sides of a square represents its boundary. Hence, the formula can be given by:

Perimeter = length of 4 sides

Perimeter = a+ a + a + a

Perimeter of square = 4a

The formula for diagonal of a square: A diagonal is a line which joins two opposite sides in a polygon. For calculating the length diagonal of a square, we make use of the Pythagoras Theorem.

In the above figure, the diagonal’ divides the square into two right angled triangles. It can be noted here that since the adjacent sides of a square are equal in length, the right angled triangle is also isosceles with each of its sides being of length ‘a’.

Hence, we can conveniently apply the Pythagorean theorem on these triangles with base and perpendicular being ‘a’ units and hypotenuse being’ units. So we have:

Solved examples:

Question 1: A square has one of its sides measuring 23 cm. Calculate its area, perimeter, and length of its diagonal.

Solution: Given,

Side of the square = 23 cm

Area of the square:

Area of the square =

Perimeter of the square:

Perimeter of the square= 4a= 4 × 23 = 92 cm

Diagonal of a square:

Diagonal of a square =

Question 2:  A rectangular floor is 50 m long and 20 m wide. Square tiles, each of 5 m side length, are to be used to cover the floor. Find the total number of tiles which will be required to cover the floor.

Solution: Given,

Length of the floor = 50 m

Breadth = 20 m

Area of the rectangular floor = length x breadth = 50 m x 20 m = 1000 sq. m

Side of one tile = 5 m

Area of one such tile = side x side = 5 m x 5 m = 25 sq. m

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.

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.

Area of Square

The area of a square is defined as the number of square units needed to fill this shape. In other words, when we want to find the area of a square, we consider the length of its side. Since all the sides of the shape are equal, its area is the product of its two sides. The common units used to measure the area of the square are square meters, square feet, square inch, and square cm.

The area of a square can also be calculated with the help of other dimensions, such as the diagonal and the perimeter of the square. Let us try to understand more about the area of the square on this page.

What is the Area of Square?

A square is a closed two-dimensional shape with four equal sides and four equal angles. The four sides of the square form the four angles at the vertices. The sum of the total length of the sides of a square is its perimeter, and the total space occupied by the shape is the area of the square. It is a quadrilateral with the following properties.

• The opposite sides are parallel.
• All four sides are equal.
• All angles measure 90º.

Squares can be found all around us. Here are some commonly seen objects which have the shape of a square. The chessboard, the clock, a blackboard, a tile, are all examples of a square.

Area of a Square Definition

The area of a square is the measure of the space or surface occupied by it. It is equal to the product of the length of its two sides. Since the area of a square is the product of its two sides, the unit of the area is given in square units.

Observe the below square shown below. It has occupied 25 squares. 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 × side = 5 × 5 = 25 square units.

Square Definition

A square is a two-dimensional shape quadrilateral with four sides equal and parallel to each other. The angles in this shape are measured as 90 degrees.

Area of a Square Formula

The formula for the area of a square when the sides are given is:

Area of a square = Side × Side = S²

Algebraically, the area of a square can be found by squaring the number representing the measure of the side of the square. Now, let us use this formula to find the area of a square of side 7 cm. We know that the area of a square = Side × Side. Substituting the length of side as 7 cm, 7 × 7 = 49. Therefore, the area of the given square is 49 cm².

The area of a square can also be found with the help of the diagonal of the square. The formula used to find the area of a square when the diagonal is given is:

Area of a square using diagonals = Diagonal²/2.

Let us understand the derivation of this formula with the help of the following figure, where ‘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’. Applying the Pythagoras theorem we have d² = s² + s²; d² = 2s²; d = √2s; s = d/√2. Now, this formula will help us to find the area of the square, using the diagonal. Area = s² = (d/√2)² = d²/2. Therefore, the area of the square is equal to d²/2.

How to Find Area of a Square?

In the above section, we covered the definition of area of square as well as area of square formula. In this section let us understand how to use the area of the square formula to find its area with the help of few applications or real-world examples.

Area of Square When the Perimeter of a Square is Given

Example: Find the area of a square park whose perimeter is 360 ft.

Solution:
Given: Perimeter of the square park = 360ft
We know that,
Perimeter of a square = 4 × side
⇒ 4 × side = 360
⇒ side = 360/4
⇒ side = 90ft
Area of a square = side²
Hence, Area of the square park = 90² = 90 × 90 = 8100 ft²
Thus, the area of a square park whose perimeter is 360 ft is 8100 ft².

Area of Square When the Side of a Square is Given

Example: Find the area of a square park whose side is 90 ft.

Solution:
Given: Side of the square park = 90ft
We know that,
Area of a square = ft²
Hence, Area of the square park = 90² = 90 × 90 = 8100 ft²
Thus, the area of a square park whose side is 90 ft is 8100 ft²

Area of Square When the Diagonal of a Square is Given

Example: Find the area of a square park whose diagonal is 14 feet.

Solution:
Given: Diagonal of the square park = 14 ft
We know that,
Area of a square formula when diagonal is given = d²/2
Hence, Area of the square park = (14 × 14)/2 = 98 ft²
Thus, the area of a square park whose diagonal is 14 m is 98 ft².

Area of Square Tips:

Note the following points which should be remembered while we calculate the area of a square.

• A common mistake that we tend to make while calculating the area of a square is doubling the number. This is incorrect! Always remember that the area of a square is side × side and not 2 × sides.
• When we represent the area, we should not forget to write its unit. 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 with a side of 3 units will have an area of 3 × 3 = 9 square units.

Area of a Square Examples

Introduction to a Square

We all are familiar with the figure, square. It is a quadrilateral wherein all the four sides and angles of it are equal. All the four angles are 90 degrees each, that is, right angles. You can also consider square as a special case of a rectangle where you will find that the two adjacent sides are of equal length.In this article, we will mainly be focusing on the various square formulas such as its area, perimeter, length of the diagonals and examples.

Area of a Square = a²

The perimeter of a Square =4a

Diagonal of a Square=a√2

Where ‘a’ is the length of a side of the square.

Properties of a Square

• The lengths of all the four sides of a square are equal.
• The two diagonals bisect each other at right angles, that is, 90°
• The lengths of diagonals of a square are equal.

Derivation of Square Formula

Derivation of Area of a Square

To better our understanding of the concept, let us take a look at the derivation of the area of a square. Let us consider a square where the lengths of its side are ‘a’ units and diagonal is ‘d’ units respectively. As you all know that area of a square is the region which is enclosed within its boundary. As we have already mentioned, that a square is a special case of a rectangle that has its two adjacent sides being of equal length. Therefore, the area can be expressed as –

Area of a rectangle = Length × Breadth

• Area of square = Length x Breadth
• Area of square = a× a = = a²

Derivation of the Perimeter of a Square

The perimeter of the square is the total length of its boundary. The boundary of a square is represented by the sum of the length of all sides. Hence, the perimeter is expressed as –

• Perimeter of a square = length of 4 sides
• Perimeter of a square = a+ a + a + a = 4a

Derivation of the Lenght of a Diagonal of a Square

As you all know that the diagonal is a line that joins the two opposite sides in a polygon. Therefore, in order to calculate the diagonal length of a square, we use the Pythagoras Theorem. If you mark a diagonal in the square, then you will realize that the diagonal divides the square into two right-angled triangles. Now, the two adjacent sides of a square are equal in length. And the right-angled triangle works as an isosceles triangle with each of its sides being of length ‘a’ units.

Thus, we can apply the Pythagoras theorem on these triangles which have base and perpendicular of ‘a’ units and hypotenuse of ‘b’ units. So, according to the formula we have:

• d²=a²+a²
• d = √2 a²
• d = a √2 units

Solved Examples

Now, that we have some understanding about the concept and meaning of square formula, let us try some examples to deepen our understanding of the topic

Example 1 – The side of a square is 5 meters. What is the area of the square in m² 

Solution 1 – As we know that the sides of а square are of equal size. And the formula for the square’s area is a × a where a is the side of the square. Hence,

Area = 5m × 5m = 25 m²

Example 2 – The perimeter of a square is 24 cm. What is the area of the square is cm²

Solution 2 – Perimeter of a square is 4 × a, where a is the side of the square
4 × a =24
a = 24 ÷ 4 = 6
Area = a × a = 6 cm × 6 cm = 36 cm²

Example 3 – The side of a square is 5 cm. If its side is doubled, how many times is the area of the new square bigger than the area of the old square?

Solution 3 – The area of the first square is 5 cm × 5 cm = 25 cms. New length = 5cm + 5 cm = 10 cm
The area of the new square is 10 cm × 10 cm = 100 cm^2. Hence, the area is 4 times bigger.

Area of Square Formula Derivation

To better our understanding of the concept, let us take a look at the derivation of the area of Square formula. Let us consider a square as a rectangular object whose length is of a unit and breadth is of a unit. As we know the area of the rectangle is given by,

A =  L × B

Where

A	Area of the square
l	Length of the rectangular object
b	The breadth of a rectangular object

A = l × b

A = a × b

= a × a = a² = a²

Solved Examples on Area of Square Formula

Now that we have some clarity about the concept and meaning of the area of the square, let us try some examples to deepen our understanding of the subject.

Q: Find the area of a square plot of side 8 m.

Ans: As we already have a formula for calculating the area of a square. Let us substitute the values

A = a × a = a² = a²

A = 8²

= 64 sq m

Q: A square of 10 cm long is cut into tiny squares of 2 cm long. Calculate the number of tiny squares that can be created.

Ans: Since the length of the big square is 10 cm, hence its Area A is:,

A= a × a = a² = a²

A = 10² = 100 cm²

Now, since the length of tiny square is 2 cm, hence its Area is:

A =  a × b

= 2 × 2

= 2 × 2 = 2²

= 4 sq cm.

Therefore, the number of squares that we can create are:

Example

Let us understand the formula much better with the help of an example.

Consider a square field having the sides of it as 23 ft. A person has to build a swimming pool over there and has to find how much area will the pool cover. Along with the pavement length needed to be done on the boundary of the pool. He is also curious to know what could be the farthest straight line length to swim in the pool. Find out the details that the person wanted.

The side given in the question is 23 ft of the pool.

So, first the area of the pool will be found with the help of formula,

Area (A)=a²

A=232=529 ft²

Now, the pavement length is the perimeter,

Therefore formula for perimeter is,

Perimeter (P)=4a

P=423=92 ft

At last the maximum straight line distance that the person could swim in a square shaped pool is the diagonal length of the pool.

Therefore, the diagonal length of the pool can be found by using formula,

d=a√2

d=23√2=32.52 ft

Hence, the area covered by the swimming pool is 529 ft2, the measure of the pavement distance needed to be done on boundary is 92 ft as well as the maximum distance that the person could swim in a straight line in his pool is 32.52 ft.

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