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What is Meant by the Area of a Square?

person holding white paper over brown wooden floorGeometry deals with the concept of flat (2-dimensional) and solid (3-dimensional) shapes and figures. The two-dimensional figures include the concept of area and perimeter while the three-dimensional figures have the concepts of surface area, lateral surface area, volume etc. The 2-d figures are basically plane figures. A closed 2-d quadrilateral with all four equal sides and all four angles equal to right angles (90 degrees) is a square. The square units used to fill a square can be termed the area of square. Generally speaking, the space occupied by any flat figure (shape) is the area of the same.

Description of a square and its properties

Any two-dimensional figure with four equal sides and angles (90 degrees) is a square shape. All the sides of the respective shape make four angles at the vertices. The sum total of sides of the square shape gives the perimeter and the space covered by the same flat shape is known as the area of the square.

Squares are common shapes found all around us. The carrom board, chessboard, blackboards etc. can be taken as examples of the shape.

Properties of a square

  • The sides opposite are parallel.
  • All sides are equal.
  • The measure of all angles is 90 degrees each.

Why is the square’s area equal to the square of its side?

  • The area of any figure depends on its dimensions, i.e., length and width. In the case of a square, all the sides are equal and so, the length and width are the same. Hence, the area of the square is the same as the square on its side.

Area of square= side*side, when the side is given.

The relation between the area of square and side can also be given as


Here is an example to understand the application of the formula.

For example: Given that a square has a side equal to 7 units. What will be the area of the given square?

Solution: Side of square= 7units

Applying the formula, A=side^2 =7^2= 49 square units.

Note: The areas for any figure are always given in square units.

The same can also be known with the help of the square’s diagonals.

If the diagonals of any square are given the relation used to calculate the area of the same will be

Area of square= (Diagonal^2)/2

Derivation of the same

If a square has a diagonal equal to d units and side is s units then,

According to the Pythagoras theorem, d^2=2s^2 i.e., d=√2s and s=d/√2.

Hence, Area of the square= (d/√2)^2= (d^2)/2.

Errors while calculating the area

  • The most common error in the application of the first area formula is that sometimes the students confuse the exponents with double sides. The approach is inappropriate as the formula has a side * side and not 2*side.
  • The units should never be left while solving the problems. The side shows a single dimension while the area represents 2-dimensions. Hence, it is given in square units.

The formulas can also help to find the side of the square if the areas are provided ,i.e.,

Side= √Area of the square.

Example: Given: Area of any square field is 4900 square units. What is the length of the side of the same?

Solution: Area of square. Field= 4900 sq. units

So, Side= √4900 = 70 sq. units.

Cuemath experts explain the concept of the area of the square in detail with well-explained examples. The app makes the understanding of the concept simple. Just like there exists a formula for the calculations of the area of the square similarly there exists a formula for the perimeter of the same. The perimeter of a square is the sum of all the sides of the square or it can be defined as 4 times the side of any square shape. The perimeter of square can also be defined as the total distance travelled while walking along the side of any square field. In case, if you are provided with the perimeter of the square then the side of the same can be easily found by reversing the relation.

The perimeter of square= 4*side of the square

Note: Perimeter is always given in units.

If the perimeter is given, the side of the square=Perimeter/4.