## AREA OF RHOMBUS

The area of a rhombus is equal to one half the product of the lengths of the diagonals. ## Formula for Area of a Square

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

Then,

Area  =  1/2 ⋅ (d1d2) sq.units

## Examples

Example 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.

Example 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.

Example 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

d2  =  192

Divide each side by 8.

d2  =  24 cm

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

Example 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

Example 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

⋅ 2x  =  48

16x  =  48

Divide each side by 16.

x  =  3

Example 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.

Example 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. Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

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