AREA OF RHOMBUS

The area of a rhombus is equal to one half the product of the lengths of the diagonals.  

Let d₁ and d₂ be the lengths of diagonals of a rhombus. 

Problem 1 :

If the lengths of the diagonals of a rhombus are  16 cm and 30 cm, find its area.

Solution :

Formula for area of a rhombus :

=   1/2 ⋅ (d₁d₂)

Substitute 16 for d₁ and 30 for d₂.

=   1/2 ⋅ (16 ⋅ 30)

=   8 ⋅ 30

=  240 cm²

So, area of the rhombus is 240 square cm.

Problem 2 :

Find the area of the rhombus shown below. 

Solution :

In the rhombus shown above,

d₁  =  5 + 5  =  10 units

d₂  =  4 + 4  =  8 units

Formula for area of a rhombus :

=   1/2 ⋅ (d₁d₂)

Substitute 10 for d₁ and 8 for d₂.

=   1/2 ⋅ (10 ⋅ 8)

=   5 ⋅ 8

=  40

So, area of the rhombus is 40 square units.

Problem 3 :

Area of a rhombus is 192 square cm. If the length of one of the diagonals is 16 cm, find the length of the other diagonal. 

Solution :

Area of the rhombus  =  192 cm²

1/2 ⋅ (d₁d₂)  =  192

Substitute 16 for d₁.

1/2 ⋅ (16 ⋅ d₂)  =  192

8 ⋅ d₂  =  192

Divide each side by 8.

d₂  =  24 cm

So, the length of the other diagonal is 24 cm. 

Problem 4 :

Area of a rhombus is 120 square units. If the lengths of the diagonals are 10 units and (7x + 3) units, then find the value of x.  

Solution :

Area of the rhombus  =  120 cm²

1/2 ⋅ (d₁d₂)  =  120

Substitute 10 for d₁ and (7x + 3) for d₂. 

1/2 ⋅ [10(7x + 3)]  =  120

5(7x + 3)  =  120

Divide each side by 5. 

7x + 3  =  24

Subtract 3 from each side. 

7x  =  21

Divide each side by 7.

x  =  3

Problem 5 :

Area of the rhombus shown below is 48 square inches. What is the value of x ? 

Solution :

In the rhombus shown above,

d₁  =  8 + 8  =  16 units

d₂  =  x + x  =  2x units

Given : Area of the rhombus is 48 square inches. 

Then, 

1/2 ⋅ (d₁d₂)  =  48

Substitute 16 for d₁ and 2x for d₂.

1/2 ⋅ (16 ⋅ 2x)  =  48

8 ⋅ 2x  =  48

16x  =  48

Divide each side by 16.

x  =  3

Problem 6 :

Find the area of the rhombus shown below. 

Solution :

Measure the lengths of the diagonals AC and BD.  

The lengths of the diagonals are  4 units and 2 units. 

Formula for area of a rhombus :

=   1/2 ⋅ (d₁d₂)

Substitute 4 for d₁ and 2 for d₂.

=   1/2 ⋅ (4 ⋅ 2)

=   2 ⋅ 2

=  4 

So, area of the rhombus is 4 square units.

Problem 7 :

Find the area of the rhombus having each side equal to 17 cm and one of its diagonals equal to 16 cm.

Solution :

Let A, B, C and D be the vertices of the rhombus. 

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 BDE. 

By Pythagorean Theorem, 

BD²  =  BE² + DE²

17²  =  BE² + 8²

289  =  BE² + 64

Subtract 64 from each side. 

225  =  BE²

15²  =  BE²

15  =  BE

Then, 

EC  =  15

Length of the diagonal BC : 

BC  =  BE + EC

BC  =  15 + 15

BC  =  30 units

So, the lengths of the diagonals are 16 units and 30 units. 

Formula for area of a rhombus :

=   1/2 ⋅ (d₁d₂)

Substitute 16 for d₁ and 30 for d₂.

=   1/2 ⋅ (16 ⋅ 30)

=   8 ⋅ 30

=  240 

So, area of the rhombus is 240 square units.

Problem 8 :

One of the diagonals of a rhombus is double and other diagonal. Its area is 25 sq.cm. The sum of the diagonal is :

a)  10 cm    b)  12 cm    c)  15 cm    d)  16 cm

Solution :

Let d₁ and d₂ be the length of diagonals.

d₁ = 2d₂

Area of rhombus =  1/2 ⋅ (d₁d₂)

25 = (1/2) ⋅ (2d₂d₂)

25 = d_2²

d₂ = √25

d₂ = 5 cm

d₁ = 2(5) ==> 10 cm

Sum of the diagonal = d₁ +  d₂

= 10 + 5

= 15 cm

So, the sum of the diagonals is 15 cm.

Problem 9 :

The perimeter of the rhombus is 56 m, and its height is 5 m. Its area is 

a)  64 sq.m    b)  70 sq.m    c)  78 sq.m    d)  84 sq.m

Solution :

Perimeter of rhombus = 56 m

4a = 56

a = 56/4

a = 14 m

Area of rhombus = base x height

= 14 x 5

= 70 sq.m

So, area of the rhombus is 70 sq.m

Problem 10 :

The length of one diagonal of a rhombus is 80% of the other diagonal. The area of the rhombus is how many times the square of the length of other diagonal ?

a)  4/5    b)  2/5    c)  3/4    d)  1/4

Solution :

Let d₁ and d₂ be the length of diagonals.

d₁ = 0.80d₂

Area of rhombus = 1/2 ⋅ (d₁d₂)

= 1/2 ⋅ (0.80d₂d₂)

= 0.40 d_2²

= (40/100) d_2²

= (2/5) d_2²

So, option b is correct.

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