A rhombus is a four-sided closed figure where the lengths of all the four sides will be equal and also the diagonals will be perpendicular.

Let 's' be the length of each side of a rhombus.

Then,

**Perimeter = 4s**

**Example 1 :**

Find the perimeter of the rhombus whose side length is 16 cm.

**Solution :**

Formula for perimeter of a rhombus :

= 4s^{ }

Substitute 16 for s.

= 4(16)

= 64

So, the perimeter of the rhombus is 64 cm.

**Example 2 :**

If the perimeter of a rhombus is 72 inches, then find the length of each side.

**Solution :**

Perimeter of the rhombus = 72 inches

4s = 72

Divide each side by 4.

s = 16

So, the length of each side of the rhombus is 16 inches.

**Example 3 :**

A rhombus has side length of 500 cm. Find its perimeter in meter.

**Solution :**

Formula for perimeter of a rhombus :

= 4s^{ }

Substitute 500 for s.

= 4(500)

= 2000 cm -----(1)

We know

100 cm = 1 m

Therefore, to convert centimeter to meter, we have to divide by 100.

(1)-----> Perimeter = 2000 cm

Divide the right side by 100 to convert cm into m.

Perimeter = (2000 / 100) m

= 20 m

So, perimeter of the rhombus is 20 meters.

**Example 4 :**

If the length of each side of a rhombus is (3x + 4) and its perimeter is 76 units, find the value of x.

**Solution :**

Perimeter of the rhombus = 76 units

4s = 76

Divide each side by 4.

s = 19

Substitute (3x + 4) for s.

3x + 4 = 19

Subtract 4 from each side.

3x = 15

Divide each side by 3.

x = 5

**Example 5 :**

In the diagram shown below, if PQRS is a rhombus, then find its perimeter.

**Solution :**

All four sides of a rhombus are congruent.

So,

RS = PS

5y - 6 = 2y + 3

Subtract 2y from each side.

3y - 6 = 3

Add 6 to each side.

3y = 9

Divide each side by 3.

y = 3

To find the length of each side of the rhombus, substitute 3 for y either in '2y + 3' or '5y - 6'.

2y + 6 = 2(3) + 3

2y + 6 = 6 + 3

2y + 6 = 9

So, the length of each side of the rhombus is 9 units.

Formula for perimeter of a rhombus :

= 4s^{ }

Substitute 9 for s.

= 4(9)

= 36

So, perimeter of the rhombus is 36 units.

**Example 6 :**

Find the perimeter of the rhombus shown below.

**Solution : **

Find the length of the side MN in the above rhombus using distance formula.

MN = √[(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}]

Substitute (x_{1}, y_{1}) = (2, 1) and (x_{2}, y_{2}) = (6, 3).

LM = √[(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}]

LM = √[(6 - 2)^{2} + (3 - 1)^{2}]

LM = √(4^{2} + 2^{2})

LM = √(16 + 4)

LM = √20

LM = 2√5

All four sides of a rhombus are congruent.

Then, the length of each side of the above rhombus is 2√5 units.

Formula for perimeter of a rhombus :

= 4s^{ }

Substitute 2√5 for s.

= 4(2√5)

= 8√5

So, perimeter of the rhombus is 8√5 units.

**Example 7 :**

In the rhombus ABCD shown below, if the lengths of the diagonals AC and BD are 10 units and 8 units respectively, find its perimeter.

**Solution : **

The diagonals of a rhombus will be perpendicular and they will bisect each other.

Then, we have

In the above rhombus, consider the right angled triangle CDE.

By Pythagorean Theorem,

CD^{2} = DE^{2} + CE^{2}

CD^{2} = 4^{2} + 5^{2}

CD^{2} = 16 + 25

CD^{2} = 41

CD = √41

All four sides of a rhombus are congruent.

Then, the length of each side of the above rhombus is √41 units.

Formula for perimeter of a rhombus :

= 4s^{ }

Substitute √41 for s.

= 4√41

So, perimeter of the rhombus is √41 units.

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