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

Let d_{1} and d_{2} be the lengths of diagonals of a rhombus.

Then,

**Area = 1/2 ⋅ (d _{1}d_{2}) sq.units**

**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 ⋅ (d_{1}d_{2})

Substitute 16 for d_{1} and 30 for d_{2}.

= ^{ }1/2 ⋅ (16 ⋅ 30)

= 8 ⋅ 30

= 240 cm^{2}

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,

d_{1} = 5 + 5 = 10 units

d_{2} = 4 + 4 = 8 units

Formula for area of a rhombus :

= ^{ }1/2 ⋅ (d_{1}d_{2})

Substitute 10 for d_{1} and 8 for d_{2}.

= ^{ }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 cm^{2}

1/2 ⋅ (d_{1}d_{2}) = 192

Substitute 16 for d_{1}.

1/2 ⋅ (16 ⋅ d_{2}) = 192

8 ⋅ d_{2} = 192

Divide each side by 8.

d_{2} = 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 cm^{2}

1/2 ⋅ (d_{1}d_{2}) = 120

Substitute 10 for d_{1} and (7x + 3) for d_{2}.

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,

d_{1} = 8 + 8 = 16 units

d_{2} = x + x = 2x units

**Given :** Area of the rhombus is 48 square inches.

Then,

1/2 ⋅ (d_{1}d_{2}) = 48

Substitute 16 for d_{1} and 2x for d_{2}.

1/2 ⋅ (16 ⋅ 2x) = 48

8 ⋅ 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 ⋅ (d_{1}d_{2})

Substitute 4 for d_{1} and 2 for d_{2}.

= ^{ }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,

BD^{2} = BE^{2} + DE^{2}

17^{2} = BE^{2} + 8^{2}

289 = BE^{2} + 64

Subtract 64 from each side.

225 = BE^{2}

15^{2} = BE^{2}

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 ⋅ (d_{1}d_{2})

Substitute 16 for d_{1} and 30 for d_{2}.

= ^{ }1/2 ⋅ (16 ⋅ 30)

= 8 ⋅ 30

= 240

So, area of the rhombus is 240 square units.

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