AREA AND PERIMETER OF A SQUARE

In this section, you will learn how to find perimeter and area of square. 

A square is a four-sided closed figure where the lengths of all the four sides will be equal and each vertex angle will be right angle or 90^o as shown below.

Example 1 :

Find the perimeter and area of a square with side 7 cm.

Solution :

Perimeter :

= 4s

Substitute s = 7.

= 4 x 7

= 28 cm

Area :

= s²

Substitute s = 7.

= 7²

= 7 x 7

= 49 cm²

Example 2 :

Find the perimeter and area of the square shown below.

Solution :

To find the perimeter and area of square, we have to know the length of side of the square.

To find the length of side of the square, let us consider the right triangle in the given square as shown below.

Use Pythagorean theorem to find the length of side of the square.

s² + s² = 5²

2s² = 25

Divide both sides by 2.

s² = 25/2

s² = 12.5

Take radical on both sides. 

√s² = √12.5

s = √12.5

Perimeter :

= 4s

Substitute s = √12.5.

= 4√12.5 cm

Area :

= s²

Substitute s = √12.5.

= (√12.5)²

= 12.5 cm²

Example 3 :

The area of a square is 32 square inches. Find the length of its diagonal.  

Solution :

Area of the square  =  32 in²

1/2 ⋅ d²  =  32

Multiply each side by 2.

d²  =  64

Find positive square root on both sides.

 √d²  =  √(8 ⋅ 8)

  d  =  8

So, the length of diagonal is 8 inches. 

Example 4 :

The square has side length 36 inches. Find its area in square feet.

Solution :

When the length of a side is given, formula for area of a square :

=  s² 

Substitute 12 for s.

=  36²

=  1296 in² -----(1)

We know  

12 inches  =  1 ft

Square both sides.

(12 inches)²  =  (1 ft)²

12² in²  =  1² ft²

144 in²  =  1 ft²

Therefore, to convert square inches into meter square feet,  we have to divide by 144. 

(1)-----> Area of the square  =  1296 in²

Divide the right side by 144 to convert in² into ft².

Area of the square  =  (1296 / 144) ft²

=  9 ft²

So, the area of the square is 9 square feet.

Example 5 :

The diagonals of two squares are in the ratio 2 : 5. Find the ratio of their areas.

Solution :

Let the diagonals of two squares be 2x and 5x respectively.

Area of a square when diagonal is given :

= (½)d²

Area of first square = (1/2)(2x)²

= (½)(4x²)

= 2x²

Area of second square = (½)(5x)²

= (½)(25x²)

= 25x²/2

Ratio of their areas is

= 2x² : 25x²/2

= 4x² : 25x²

=  4 : 25

Note : 

When the ratio of lengths of sides of two squares is given, to find the ratio of their areas, we have to square the lengths of sides and take ratio. 

Example 6 :

A square is of area 64 cm². What is its perimeter ?

Solution :

Area of a square = 64 cm²

s² = 64

Take square root on both sides. 

s = √64

s = 8 cm

Perimeter of the square :

= 4s

Substitute s = 8.

= 4(8)

= 32 cm

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