Area of Square: A square is a 2-D plane shape that has four internal angles subtended by its four sides. In the square, all sides are the same in measure and all four internal angles are 90-degree angles. A square is defined as a quadrilateral shape in geometry having four equal sides in length and parallel to each other. In our surroundings, you can notice several square-designed stuff like paper napkins, chess boards, bread, cushions, clock, stamps, tiles, etc. In this article, we shall discuss the area of a square, its formulas, and some solved examples for a clear understanding of the concepts of the square.

What is the Area of Square?

In the study of geometry, the area of a square refers to the total number of square units required to fill a particular square shape. It is the measure of the surface covered by the square shape. It is equivalent to the multiplication of the length of 2 sides of a square. As the area of a square is produced by the multiplication of its 2 sides, the unit of the area of the square is expressed in square units. Look at the following square as shown below. It has covered the 25 small squares. Hence the area of the square is 25 square units. From the given figure, you can find that the length of the side of the square is 5 units. Hence the area of the given square will be = side × side = 5 × 5 = 25 square units. The measurement for finding the square area is done in the standard unit like square meters (m²).

Area of Square Formula

The formula for calculating the area of a square in geometry is explained below if the sides of a square are denoted by ‘a’.

Area of a Square = Side × Side = a²

Mathematically, the area of a square can be expressed by squaring the value of the sides of the square as . For example, the area of a square of side 3 cm is 3²= 9 cm². 

The formula of the area of a square can also be expressed by using the diagonal length of the square. The area of a square formula is explained below if the diagonal of that square is given in the question.

Area of a Square using Diagonals = Diagonal² / 2

You can derive the above formula by using the following figure, where ‘d’ is diagonal and ‘s’ is the side of the square. Using the Pythagoras theorem we get d² = s² + s²; d² = 2s²; d = √2s; s = d/√2. 

Now, the resultant expression will help to find the area of the square, using the diagonal. 

Area = s² = (d/√2)² = d²/2. So the area of the square is equivalent to d² / 2.

Area of Square Examples

Question 1: Calculate the area of a square cushion whose side is 9 cm.

Solution: Given that the side of the square cushion = 9 cm

As we know that the area of a square = a²

So Area of the square cushion = 9² = 9 × 9 = 81 cm²

Therefore, the area of a square-shaped cushion with a side of 9 cm is 81 cm².

Question 2: Calculate the area of a square stamp whose diagonal is 1.5 cm.

Solution: Given that the diagonal of the square stamp = 1.5 cm

As we know that the area of a square = d²/ 2 

So Area of the square stamp = (1.5)² / 2 = 2.25 / 2 = 1.125 cm²

Therefore, the area of a square-shaped stamp with a side of 1.5 cm is 1.125 cm².

Question 3: Calculate the value of the area of a square-shaped plot whose one side is 12m.

Solution: Given that the side of the square-shaped plot = 12 m

As we know that the area of a square = a²

So Area of the square-shaped plot = 12² = 12 × 12 = 144 m²

Therefore, the area of a square-shaped plot with a side of 12 m is 144 m².

Question 4: The area of square-shaped chess is 1600 cm². Find the value of the length of its side.

Solution: Given that the area of the chess = 1600 cm²

As we know that the area of a square = a²

So the length of the square-shaped chess = √Area = √1600 = 40 cm

Therefore, the length of a square-shaped chess board with its area of 1600 cm² is 40 cm.

Question 5: Calculate the area of a square floor whose diagonal is 11 m.

Solution: Given that the diagonal of the square floor = 11 m

As we know that the area of a square floor = d²/ 2 

So Area of the square floor = (11)² / 2 = 121 / 2 = 60.5 m²

Therefore, the area of a square-shaped floor with a side of 11 m is 60.5 m².

Area of Square: FAQs

Ans. In the study of geometry, the area of a square refers to the total number of square units required to fill a particular square shape.

Ans. The area of a square formula is equivalent to the multiplication of the length of 2 sides of a square. Area of a Square = Side × Side = a²

Ans. The area of a square formula if the diagonal of that square is given in the question will be Area of a Square using Diagonals = Diagonal² / 2

Ans. The standard unit for the area of a square is square units like m².

Ans. A square is defined as a quadrilateral shape in geometry having four equal sides in length and parallel to each other.

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