# 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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