PERIMETER AND AREA OF 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 90as shown below.

Formula for Perimeter of Square

The distance around a two dimensional shape is called perimeter. 

If s be the length of each side of a square, then the perimeter of the square is 

=  s + s + s + s

=  4s

So, the formula for perimeter of a square :

Perimeter  =  4s

Formula for Area of Square

The amount of space available inside the boundary of a two-dimensional space is called area. 

We can use the following two formulas to calculate the amount of space available inside the square.  

If s be the length of each side of a square, then the formula for area of a square :

Area  =  s2

If d be the length of each diagonal of a square, then, the formula for area of a square : 

Area  =  1/2 ⋅ d2

Example 1 :

If the length of each side of a square is 8.5 cm, then find its perimeter. 

Solution :

Formula for perimeter of a square :

=  4s 

Substitute 14 for s.

=  4(8.5)

=  34

So, the perimeter of the square is 34 cm.

Example 2 :

The length of each diagonal of a square is 2√2 cm. Find its perimeter.

Solution :

To find the perimeter of a square, first we have to know the length of each side.

Let s be the length of each side of the square. 

Draw a sketch. 

In the figure shown below, consider the right triangle ABC. 

By Pythagorean Theorem, we have

AB2 + BC2  =  AC2

Substitute.

s2 + s2  =  (2√2)2

Simplify and solve for s. 

2s2  =  22 (√2)2

2s2  =  4 (2)

2s2  =  8

Divide each side by 2. 

s2  =  4

Find positive square root on both sides.

 √s2  =  √4

 √s2  =  √(2 ⋅ 2)

s  =  2

Formula for perimeter of a square. 

Perimeter  =  4s

Substitute 2 for s. 

=  4(2)

=  8

So, the perimeter of the the square is 8 cm.

Example 3 :

If a square has the side length of 7.5 cm, then find its area.

Solution :

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

=  s

Substitute 24 for s.

=  (7.5)2

=  56.25

So, area of the square is 56.25 square cm.

Example 4 :

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 5 :

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 :

=  s

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. 

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