## Area of Square

The area of a square is defined as the number of square units needed to fill a square. 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 a square 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, 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 in 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 square 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.

### Definition of Area of a Square

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 green 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.

## 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 = S ^{2}**. 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 7 cm, 7 × 7 = 49. Therefore, the area of the given square is 49 cm

^{2}.

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 = Diagonal ^{2}/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 d^{2} = s^{2} + s^{2}; d^{2} = 2s^{2}; d = √2s; s = d/√2. Now, this formula will help us to find the area of the square, using the diagonal. Area = s^{2} = (d/√2)^{2} = d^{2}/2. Therefore, the area of the square is equal to d^{2}/2.

Before getting to the area of square formula used for calculating the region occupied, let us try using a 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 = Side ^{2 }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 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,

## 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.

### Find 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 = side^{2}

Hence, Area of the square park = 90^{2} = 90 × 90 = 8100 ft^{2}

Thus, the area of a square park whose perimeter is 360 ft is 8100 ft^{2}.

### Find 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 = ft^{2}

Hence, Area of the square park = 90^{2} = 90 × 90 = 8100 ft^{2}

Thus, the area of a square park whose side is 90 ft is 8100 ft^{2}

### Find 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 = d^{2}/2

Hence, Area of the square park = (14 × 14)/2 = 98 ft^{2}

Thus, the area of a square park whose diagonal is 14 m is 98 ft^{2}.

**Important Notes on Area of Square**

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

**Example 1:** What is the area of a square-shaped swimming pool whose one side is equal to 8 m?

**Solution:**

We know that one side of the swimming pool is 8 m, so, we will use the formula: Area of a square = side × side = 8 × 8 = 64 m^{2}. Therefore, the area of the swimming pool is 64 square meters.

**Example 2: **The area of a square-shaped carrom board is 3600 cm^{2}. What is the length of its side?

**Solution:**

Area of the square carrom board = 3600 cm^{2}. We know that Area = side × side = side^{2}. So, side = √Area = √3600 = 60 cm. Therefore, the side of the carrom board is 60 cm.

**Example 3: **Find the area of the square-shaped floor room which is made up of 100 square tiles of side 15 inches.

**Solution:**

Area of one tile = 15 inch × 15 inch= 225 square inches. We know that there are 100 tiles on the floor of the room. Thus, the area occupied by 100 tiles is the floor area = 100 × 225 square inches = 22500 square inches. Therefore, the area of the floor is 22500 square inches.

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

**Solution:**

The area of a square when its diagonal is given is D^{2}/2. Given, diagonal d = 4 ft. Area of the carpet = (4 × 4)/2 = 16/2 = 8 square feet. Therefore, the area of the carpet is 8 square feet.

### 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 = 5625 sq. m

For 1 sq. m, the cost of painting = Rs. 3

Thus, for 5625 sq. m, the cost of painting = Rs. 3 × 5625 = Rs 16875

**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

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 1: **Find the area of a square of side length 15 m

__Solution__:

Area of a square = a^{2} = 15^{2} = 225 m^{2}

**Example 2: **Calculate the area of square, where the square has 35cm side’s length.

__Solution__:

Area of square is defined by a × a.

Area = 35 × 35

Area = 1225cm

**Example 3: **What is the area of a square field, if its perimeter is 32 yd?

__Solution__:

The perimeter of the square field = 32 yd and since the perimeter of a square is given by P = 4s, where s is the length of the side. We can easily determine the length by isolating s from the formula above:

s = P/4 = 32 / 4 = 8 yd

The area of the square field = s × s

Substitute the value of s, we have:

Area = 8 × 8 = 64 yd^{2}

The area of the square field is therefore 64 yd^{2}.

**Example 4: **The side of a square park is 200 m. What will be the cost of grassing it @ $0.5 per sq m?__Solution__:

** **What we need to do, is to find the area of the park then multiply the area by the cost per m^{2}.

Area of the square park = side × side

*A = s²*

Substitute values and simplify.

A = 200 × 200

A = 40,000 m^{2}

Area of grassing = area of park = 40,000 sq m.

Cost of grassing = area of grassing × rate per sq meter.

Substitute values we will get:

Cost = 40,000 x 0.5 = $20,000

Therefore, Cost of grassing is $20,000.

**Example 5: **A square lawn is surrounded by a path 2 m wide around it. If the area of the path is 160 sq m, find the area of the lawn.

__Solution__**:**

Given: A square lawn is surrounded by 2 m wide path; area of path is 160 sq m.

To find: Area of lawn.

(Hint: Lawn is surrounded by the path i.e the path is at external edge of lawn. to find area of lawn subtract area of paths from total area)

Let side of the lawn be y, then we have:

External side including path = side of lawn + width of path on both sides.

= y + (2 + 2)

= y + 4

Total area including path = (y + 4) × (y + 4).

= y² + 8y + 16 (i).

And the area of lawn = (side)² = y × y = y² (ii).

Since the area of the path is given (160 m^{2}), we have:

Area of path = Total area including path – area of lawn.

A = (i) – (ii).

Substitute the given values we will the following equation and by isolating y, we can determine the length of side of the lawn:

160 = (y² + 8y + 16) – y²

160 = y² + 8y + 16 – y²

160 = y² – y² + 8y + 16

160 = 8y + 16

160 – 16 = 8y

144 = 8y

18 = y

Side of lawn = 18 m

Area of the lawn = side × side

A = s²

A = 18 × 18

A = 324 m^{2}

Hence the area of lawn = 324 m^{2}.

## FAQs on Area of Square

### What is Area of Square in Geometry?

In geometry, the square is a shape with four equal sides. The area of a square is defined as the number of square units that make a complete square. It is calculated by using the area of square formula Area = s × s = s^{2} in square units.

### What Is the Formula For Finding the Area of a Square?

When the side of a square is known, the formula used to find the area of a square with side ‘s’: Area = s × s = s^{2}. When the diagonal ‘d’ of the square is given, then the formula used to find the area is, Area = d^{2}/2.

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

The area of a square is calculated with the help of the formula: Area = s × s, where, ‘s’ is one side of the square. Since the area of a square is a two-dimensional quantity, it is always expressed in square units. For example, if we want to calculate the area of a square with side 4 units, it will be: A = 4 × 4 = 16 unit^{2}. Check now area of square calculator for quick calculations.

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

The perimeter of a square is a sum of four sides of a square that is P = 4 × Sides. It is given in terms of m, cm, ft, inches.

The area of square = Area = s × s, where, ‘s’ is one side of the square. It is given in terms of m^{2}, cm^{2}, ft^{2}, in^{2}.

### How to Find the Area of a Square From the Diagonal of a Square?

The area of a square can also be found if the diagonal of the square is given. The formula that is used in this case is: Area of a square using diagonals = Diagonal²/2. For example, the diagonal of a square is 6 units, the Area = 6²/2 = 36/2 = 18 square units.

### How to Find the Area of a Square From the Perimeter of the Square?

The area of a square can be calculated if the perimeter of the square is known. Since the perimeter of a square is: P = 4 × side, we can find the side of the square ‘s’ = Perimeter/4. After getting the side, the area of a square can be calculated with the formula: A = s × s. For example, if the perimeter of a square is 32 units, we will substitute this value in the formula: P = 4 × side. 32 = 4 × side. So, the side will be 8 units. Now, we can calculate the area of the square with side 8 units. Area = s × s = 8 × 8 = 64 square units.

### What Are the Units of the Area of a Square?

Since the area of a square is a two-dimensional shape, it is always expressed in square units The common units of the area of a square are m^{2}, inches^{2}, cm^{2}, foot^{2}.

### What Is the Area of a Square Inscribed In a Circle?

If a square is inscribed in a circle, the diagonal of the square is equal to the diameter of the circle. So, if the diameter of the circle is given, this value can be used as the diagonal of the square, and the area of the square can be calculated with the formula: Area of a square using diagonals = Diagonal²/2.

### 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 diagonal is given?

If the diagonal of a square is given, then the formula to calculate the area of a square is:

A = (½) × d^{2} 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.

## Area of Plane Shapes

A harder example:

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