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How to Find the Area of a Square?

area of square

The area of a square can be understood by how much space a square covers inside it. In simple terms, the space present within the boundary of a square is known as the area of the square. In this article, you shall learn the fundamental parameters of a square. Also, you will study how to find the area of a square, the area of the square formula, and the surface area of a square pyramid. 

Are we all familiar with what a square is? A square is a closed quadrilateral. Quadrilaterals are figures having 4 sides. Thus square is a four-sided figure which has all four sides equal. If one side of a square is 10 cm, then the other sides are also equal to 10 cm. Let us learn some of the mathematical terms and concepts related to a square first:

  • A square has all sides equal. This implies that the opposite and the adjacent sides of a square are equal to each other.
  • The opposite sides of a square are parallel, making it a parallelogram.
  • The adjacent sides of a square are perpendicular to each other. This means that any two adjacent sides have an angle of 90 degrees between them.
  • A square is divided into two right-angled congruent triangles. 
  • A square is a special case of a rectangle.
  • The perimeter of a square: The distance covered by the boundaries of a square is known as the perimeter of a square. It is formulated as:

Perimeter (square) = s + s + s + s = 4 x s = 4s {where ‘s’ represents the side of a square}

In our day-to-day life, we can find squares everywhere. From our homes to our schools, squares are present at each corner. The tiles in your kitchen are square. The chessboard is a square containing 64 black and white smaller squares. The most common example is a Rubrik’s cube. Each surface of a Rubik’s cube is square. 

Other dimensions, such as the diagonal and the perimeter of the square, can also be used to compute the area of a square. In this article, we’ll try to learn more about the area of the square.

What is the Area of Square?

Now we are crystal clear about what a square is! Read the section above to clarify all your doubts if you are still unsure. Let us learn what area is? The space swept or covered by any closed figure is the area. The area of a figure is defined as the space inside the figure’s boundary. As a result, the area of the square is defined as the region swept by or covered by a square. In other terms, the area of a square is the two-dimensional space within its border.

We can also say that the amount of tiny squares of dimensions 1 unit which can fill the square is known as the area of a square. Let us understand this concept with the help of the illustration given below.

Illustration: Let us consider a square of length 4 units. Now consider smaller squares of length 1 unit each. As we can see in the figure below, 4 squares of 1 unit fill the first row of the larger square. Similarly, 4 squares of 1 unit fill the second, third and fourth row. Now the larger square is filled. If we count the number of smaller squares, we get that 16 squares of 1 unit fill the square of 4 units. Hence 16 units are the area of the square. 

From this, we can deduce that the area of a square is equal to the product of its two sides. As we know, 16 = 4 x 4, and 4 units form one side of the square. In the next section, we will learn and derive the area of a square formula. 

Area of a Square Formula

From the above illustration, we learn that the area of a square is equal to the product of the sides. This can be written as ‘side x side’. Hence the formula for any square with any length of a side is given as 

Area = (Side)2

Let us look at an example to understand this formula:

Example: Find the area of a square with sides of length 13 cm.

Solution: Given the length of the side = 13 cm

                 The area of square = (Side)2 = (13)2 = 169 cm2.

The area of a square is always in square units ( square cm, square m, square inches, etc.)

What if we are not provided with the sides of a square but rather we are given the diagonal length? How do we find the area of a square in this case? 

Don’t worry! The area of the square can be evaluated even if the length of the diagonal is given. You can find the area using the formula written below:

Area of square ( using diagonals) = (D)2/2, where D represents the diagonal length.

Note: Remember that the square’s diagonals are equal, so the area remains the same if any of the diagonals are given. 

Example: Find the area of a square when the diagonal length is 13 cm.

Solution: We are given, diagonal = 13 cm

                Area of the square = D2/2

                                               = (13)2/2 

                                               = 169/2 cm2

How Do You Find The Area Of A Square

Hitherto, we have learned 2 formulas related to finding the area of a square. Let us learn how you will approach the questions related to the area of any square. 

  1. When anyone side is given:
  • Step 1: Note down the value of the side, say ‘a’.
  • Step 2: Substitute the value of a in the formula -> Area (with side) = (Side)2 = (a)2
  • Step 3: Write the answer in square units. 

Example: Find the area of plastic required to cover a square table of length 8 m.

Solution: Given that length of table = 8 m

                 Therefore the area of plastic required to cover the table = area of the table.

                  Area of table = (side)2 = 82 = 64 m2

  1. When any one diagonal is given:
  • Step 1: Write down the value of the diagonal length, say ‘d’.
  • Step 2: Substitute the value of d in the formula -> Area (with diagonal) = (d)2/2 = 
  • Step 3: Write the result in square units. 

Example: Find the area of a square with a diagonal length of 4 cm

Solution: Given that length of diagonal = 4 cm

                  Area of the square = (4)2/2 = 16/2 = 8 cm2

Find Area of Square When the Perimeter of a Square is Given

In the above sections, we learned how to calculate the area of a square when either the side or diagonal is given. But, suppose you are not provided with any of these parameters, but the perimeter of the square is given. How will you find the area when the perimeter of a square is given? Let us find out:

  • Step 1: Note the perimeter of the given square. 
  • Step 2: We know the value of the perimeter of a square is 4s. Therefore 4s = Perimeter.
  • Step 3: Substitute the value of perimeter and find the side using the formula s = Perimeter/4
  • Step 4: Now that we know the side of the square. Find the area using s2.
  • Step 5: Write the answer in square units.

Example: A square garden has a perimeter of 64 cm. Max wants to plant flowers and find the area of this garden but doesn’t know how to do it? Help him figure out the area of the garden.

Solution: We know: Perimeter of the garden = 64 cm

First, we will figure out the length of the sides of the garden.

Using the formula in step 3 

Side (s) = perimeter/4

              = 64/4

              = 16 cm 

Now,

Area of the garden = (s)2 

                               = 162

                               = 256 cm2

Some tips from our side:

Take note of the following factors to keep in mind when you calculate the area of a square.

  • While evaluating the area of a square, we often mistake doubling the number. This is not the case! Keep in mind that the area of a square is not 2 x side. It is always either ‘side x side’ or side2.
  • We must remember to write the area’s unit when representing it. The area of a square is always two-dimensional; hence we use square units. For example cm2, m2, inch2, etc.

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