Area of a Square Formula

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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 = 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.

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 = 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 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 = 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.

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 = 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

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 = 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.

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

Area of Plane Shapes

A harder example:

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