# Area of Square

Area of Square
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In terms of geometry, the area of a square is the space occupied by it. It 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 a 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 = 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 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.

### Common Mistakes

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 × side.

• 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 side 3 units will have an area of 3 × 3 = 9 square units.

### Important Topics

Given below is the list of topics that are closely connected to the area of a square. These topics will also give you a glimpse of how such concepts are covered in Cuemath.

## FAQs on Area of a Square

### 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 = s2. When the diagonal 'd' of the square is given, then the formula used to find the area is: Area = d2/2.

### 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 unit2.

### 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 quantity, it is always expressed in square units The common units of the area of a square are m2, inches2, cm2, foot2

### 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.

## Solved Examples on Area of a Square

Example 1: What is the area of a swimming pool which is in the shape of a square, with one side 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 m2. Therefore, the area of the swimming pool is 64 square meters.

Example 2: The area of a square-shaped carrom board is 3600 cm2. What is the length of its each side?

Solution:

Area of the square carrom board = 3600 cm2. We know that Area = side × side = side2. So, side = √Area = √3600 = 60 cm. Therefore, the side of the carrom board is 60 cm.

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

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 D2/2. 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.

## Practice Questions

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

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