Area of a Square: Formulas, Definition, and Solved Examples

Area of a Square: A square is a two-dimensional geometric plane shape characterized by four equal sides and four internal 90-degree angles. By definition, a square is a special type of quadrilateral with equal side lengths that are parallel to one another. You can find examples of squares in everyday items such as chessboards, tiles, napkins, and stamps. In this article, we will explore the area of a square, its essential formulas, and provide solved examples to help you master these geometric concepts.

What is the Area of Square?

In geometry, the area of a square represents the total surface space covered within its boundaries, measured in square units. This area is calculated by multiplying the length of two sides of the square. Since the area is derived from multiplying two side lengths, the unit of measurement is always expressed in square units. For instance, if a square covers 25 small unit squares, its area is 25 square units. Given a side length of 5 units, the calculation is: side × side = 5 × 5 = 25 square units. Standard scientific measurements typically use square meters (m²).

Area of Square Formula

If the side length of a square is represented by ‘a’, the standard formula for calculating its area is as follows:

Area of a Square = Side × Side = a²

Mathematically, you can find the area by squaring the length of one side (a²). For example, a square with a side length of 3 cm has an area of 3² = 9 cm². 

The area of a square can also be calculated using its diagonal length. This alternative method is particularly useful when the diagonal measurement is provided instead of the side length.

Area of a Square using Diagonals = Diagonal² / 2

You can derive this formula using the Pythagorean theorem, where ‘d’ is the diagonal and ‘s’ is the side length. Since d² = s² + s², we get d² = 2s². Consequently, d = √2s, which means s = d/√2. 

By substituting this expression for 's', we can easily solve for the area using only the diagonal measurement. 

Area = s² = (d/√2)² = d²/2. Therefore, the area of a square is equal to half the square of its diagonal.

Area of Square Examples

Question 1: Find the area of a square cushion with a side length of 9 cm.

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

Using the formula, area of a square = a².

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

Therefore, the area of the square-shaped cushion is 81 cm².

Question 2: Find the area of a square stamp with a diagonal of 1.5 cm.

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

Using the formula, area of a square = d²/2. 

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

Therefore, the area of the square-shaped stamp is 1.125 cm².

Question 3: Calculate the area of a square plot with a side length of 12 m.

Solution: Given: side of the square plot = 12 m.

Using the formula, area of a square = a².

Area of the square plot = 12² = 12 × 12 = 144 m².

Therefore, the area of the square plot is 144 m².

Question 4: A square-shaped chessboard has an area of 1600 cm². Find its side length.

Solution: Given: area of the chessboard = 1600 cm².

Since area = a², then side length (a) = √Area.

Side length = √1600 = 40 cm.

Therefore, the side length of the chessboard is 40 cm.

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

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

Using the formula, area of a square = d²/2.

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

Therefore, the area of the square floor is 60.5 m².