**Perimeter and Area of Square :**

In this section, we are going to 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.

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**

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 = s ^{2}**

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

**Area = 1/2 ****⋅ d ^{2}**

**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

AB^{2} + BC^{2} = AC^{2}

Substitute.

s^{2} + s^{2} = (2√2)^{2}

Simplify and solve for s.

2s^{2} = 2^{2} ⋅(√2)^{2}

2s^{2} = 4 ⋅(2)

2s^{2} = 8

Divide each side by 2.

s^{2} = 4

Find positive square root on both sides.

√s^{2} = √4

√s^{2} = √(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^{2 }

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 in^{2}

1/2 ⋅ d^{2} = 32

Multiply each side by 2.

d^{2} = 64

Find positive square root on both sides.

√d^{2} = √(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^{2 }

Substitute 12 for s.

= 36^{2}

= 1296 in^{2} -----(1)

We know

12 inches = 1 ft

Square both sides.

(12 inches)^{2} = (1 ft)^{2}

12^{2} in^{2} = 1^{2} ft^{2}

144 in^{2} = 1 ft^{2}

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

(1)-----> Area of the square = 1296 in^{2}

Divide the right side by 144 to convert in^{2} into ft^{2}.

Area of the square = (1296 / 144) ft^{2}

= 9 ft^{2}

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

After having gone through the stuff given above, we hope that the students would have understood, "Perimeter and Area of Square".

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