PERIMETER OF RHOMBUS

In this page perimeter of rhombus, we are going to see formulas to be used to find the perimeter of the shape rhombus.

Ways to find perimeter of rhombus:

We have three ways to find perimeter of the shape rhombus.

(i) Using side length

(ii) Using length of diagonals

(iii) Using one diagonal and one angle

Perimeter using side length:

In rhombus length of all sides will be equal. So, perimeter of rhombus is nothing but sum of all sides.

Perimeter  =  a + a + a + a  =  4 a 

here "a" stand for length of each side.

Perimeter using length of diagonals :

Perimeter = 2 √ d₁² + d₂²

here  d₁ and d₂ stands for length of diagonals.

Perimeter using one diagonal and one angle :

Let us see an example problem to understand how to find perimeter of the shape rhombus using one diagonal and one angle.

Example 1:

Find the perimeter of the shape rhombus given below.

Solution :

According to the properties of rhombus

(i) Each diagonals bisect the vertex angle.

(ii) The diagonals of rhombus are perpendicular and they bisect each other.

Here AC = 16 cm

So, AE = 8 cm  ∠EAB = 35° and ∠AEB = 90°

In triangle AEB, 

AB = hypotenuse side, BE = opposite side and AE = adjacent side

cos θ = Adjacent side/hypotenuse side

 cos 35° = AE / AB

0.819  =  8 / AB 

AB = 8 / 0.819

=  9.768

perimeter of rhombus ABCD = 4(9.768)  =  39.07 cm

Hence, perimeter of rhombus = 39 cm

Example 2:

Find the length of the other diagonal and area of rhombus.If the perimeter of a rhombus is 20 cm and One of the diagonals is of length 8 cm.

Solution:

Since one of the diagonal is given and we have to find the other diagonal. So we have to use the first formula.

Perimeter of the rhombus = 2 √ d₁² + d₂²

Here d₁ = 8 cm

Perimeter of the rhombus = 20 cm

2 √8² + d₂²  =  20 ==> √64 + d₂²   =  20/2 ==>  64 + d₂² = (10)²

64 + d₂² = 100 ==> d₂² = 36 ==> d₂ = 6 cm

So area of rhombus = (1/2) x 8 x 6

 = 4 x 6==> 24 cm²

Example 3:

Find the length of each side of rhombus if the perimeter is measuring 260 cm.

Solution:

Perimeter of the rhombus = 260 cm

                              4a = 260 m

                                a = 260/4

                                a = 65 cm

Example 4:

Find the perimeter of the rhombus whose vertices are A (2,-3), B (6,5), C (-2,1) and D (-6,-7).

Solution:

To find the perimeter of the rhombus first we have to find the length of the diagonals.

Distance between two points = √(x₂ - x₁)² + (y₂ - y₁)² 

Length of diagonal AC

A (2,-3) C (-2,1) 

x₁ = 2      y₁ = -3

x₂ = -3     y₂ = 1

                            =  √(-3 - 2)² + (1 - (-3))²

                            =  √(-5)² + (1 + 3))²

                            =  √25 + (4))²

                            =  √(25 + 16)

                            = √41

Length of diagonal BD

B (6,5) D (-6,-7) 

x₁ = 6      y₁ = 5

x₂ = -6     y₂ = -7

                            =  √(-6 - 6)² + (-7 - 5)²

                            =  √(12)² + (12)²

                            =  √144 + 144

                            =  √288

Perimeter of rhombus = 2 √ (√41)² + √288)²)

                             = 2 √ 41 + 288

                             = 2 √329 cm

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