AREA AND PERIMETER OF A SQUARE

Area :

Area of a square is defined as the space occupied by the square shaped object on a flat surface. The area of a square can be measured by comparing the shape to squares of a fixed size.

Perimeter :

Perimeter of a square is a path that surrounds the square. The term may be used either for the path or its length it can be thought of as the length of the outline of a square.

Formulas

Formulas for the perimeter P and area A of a square :

side length = s

Perimeter = 4s

Area = s²

The measurements of perimeter and circumference use units such as centimeters, meters, kilometers, inches, feet, yards, and miles. The measurements of area use units such as square centimeters (cm²), square meters(m²), and so on.

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 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 :

= (1/2)d²

Area of first square = (1/2)(2x)²

= (1/2)(4x²)

= 2x²

Area of second square = (1/2)(5x)²

= (1/2)(25x²)

= 25x²/2

Ratio of their areas is

= 2x² : 25x²/2

=  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 4 :

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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