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.
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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