Area of Rhombus
The rhombus is a quadrilateral whose all sides are equal, and diagonals intersect each other at 90 degrees. Rhombus is a diamond shape quadrilateral. Some also call the rhombus a special kind of parallelogram.
In mathematics, there are a lot of questions regarding a rhombus.
Rhombus formula
|
When both diagonals are given |
A = ½ × d1 × d2 |
|
When base and height are given |
A = b × h |
|
When angle is given |
A = b2 × Sin(a) |
Area Formula
The area \( A \) of a rhombus can be calculated using the following formulas:
\( A = \frac{d_1 \cdot d_2}{2} \)
\( A = a^2 \sin(\theta) \)
- Using the lengths of the diagonals:
- Using the length of a side \( a \) and an angle \( \theta \):
Example Calculations
Example 1
Find the area of a rhombus with diagonals measuring 10 cm and 24 cm.
Solution:
\[ A = \frac{d_1 \cdot d_2}{2} = \frac{10 \cdot 24}{2} = 120 \text{ cm}^2 \]
Example 2
Calculate the area of a rhombus with a side length of 5 cm and an angle of 60 degrees.
Solution:
\[ A = a^2 \sin(\theta) = 5^2 \sin(60^\circ) = 25 \cdot \frac{\sqrt{3}}{2} \approx 21.65 \text{ cm}^2 \]
Example 3
A rhombus has an area of 48 m⊃2; and one diagonal measures 8 m. Find the length of the other diagonal.
Solution:
\[ 48 = \frac{8 \cdot d_2}{2} \implies d_2 = 12 \text{ m} \]
Finding area of a Rhombus:
If “b” is its sides and “θ” is an included angle, the formula is: Area of a Rhombus = b2 sin θ square units.
Example: The length of a side of a rhombus is 9cm, and the included angle is 30 degrees. Then the area should be 92 sin 30 square units. The area is 40.5 square units.
There are other ways to find the area of a rhombus too.
One of the ways is by using the formula D1 x D2/2. D1 and D2 are the values of the diagonal.
Example: The length of Diagonal 1 is 6cm, and the length of Diagonal 2 is 8cm. Putting it into the formula should give us 6 x 8/2, which are 24 units squared.
Area of Rhombus Word Problems
Problem 1
A rhombus has diagonals measuring 10 cm and 24 cm. What is its area?
Solution:
The area \( A \) of a rhombus can be calculated using the formula:
\[ A = \frac{d_1 \cdot d_2}{2} = \frac{10 \cdot 24}{2} = 120 \text{ cm}^2 \]
Problem 2
A rhombus has an area of 48 m⊃;2; and one diagonal measuring 8 m. What is the length of the other diagonal?
Solution:
Using the area formula:
\[ A = \frac{d_1 \cdot d_2}{2} \implies 48 = \frac{8 \cdot d_2}{2} \implies d_2 = 12 \text{ m} \]
Problem 3
The diagonals of a rhombus are in the ratio of 3:5. If the shorter diagonal measures 9 cm, what is the area of the rhombus?
Solution:
Let the diagonals be \( d_1 = 9 \) cm and \( d_2 = 15 \) cm.
\[ A = \frac{d_1 \cdot d_2}{2} = \frac{9 \cdot 15}{2} = 67.5 \text{ cm}^2 \]
Problem 4
A rhombus has each side measuring 5 cm. If one angle measures 60 degrees, what is the area of the rhombus?
Solution:
Using the formula for area:
\[ A = a^2 \sin(\theta) = 5^2 \sin(60^\circ) = 25 \cdot \frac{\sqrt{3}}{2} \approx 21.65 \text{ cm}^2 \]
Problem 5
A rhombus has an area of 72 m⊃;2; and one diagonal measuring 16 m. Find the length of the other diagonal.
Solution:
\[ A = \frac{d_1 \cdot d_2}{2} \implies 72 = \frac{16 \cdot d_2}{2} \implies d_2 = 9 \text{ m} \]
Problem 6
The diagonals of a rhombus are 14 cm and 48 cm. What is its area?
Solution:
\[ A = \frac{d_1 \cdot d_2}{2} = \frac{14 \cdot 48}{2} = 336 \text{ cm}^2 \]
Problem 7
A rhombus has an area of 45 cm⊃;2; and one diagonal measuring 15 cm. What is the other diagonal?
Solution:
\[ A = \frac{d_1 \cdot d_2}{2} \implies 45 = \frac{15 \cdot d_2}{2} \implies d_2 = 6 \text{ cm} \]
Problem 8
A rhombus has each side measuring 10 m. If one of its angles is 30 degrees, calculate the area.
Solution:
\[ A = a^2 \sin(\theta) = 10^2 \sin(30^\circ) = 100 \cdot \frac{1}{2} = 50 \text{ m}^2 \]
Problem 9
A rhombus has diagonals of lengths 12 cm and 16 cm. What is its area?
Solution:
\[ A = \frac{d_1 \cdot d_2}{2} = \frac{12 \cdot 16}{2} = 96 \text{ cm}^2 \]
Problem 10
The area of a rhombus is 60 m⊃;2;, and one diagonal is 10 m. Find the length of the other diagonal.
Solution:
\[ A = \frac{d_1 \cdot d_2}{2} \implies 60 = \frac{10 \cdot d_2}{2} \implies d_2 = 12 \text{ m} \]
Problem 11
A rhombus has diagonals measuring 20 cm and 30 cm. What is its area?
Solution:
\[ A = \frac{d_1 \cdot d_2}{2} = \frac{20 \cdot 30}{2} = 300 \text{ cm}^2 \]
Problem 12
A rhombus has an area of 100 m⊃;2; and one diagonal measuring 20 m. What is the other diagonal?
Solution:
\[ A = \frac{d_1 \cdot d_2}{2} \implies 100 = \frac{20 \cdot d_2}{2} \implies d_2 = 10 \text{ m} \]
Problem 13
A rhombus has sides of length 8 cm and one angle measuring 45 degrees. What is its area?
Solution:
\[ A = a^2 \sin(\theta) = 8^2 \sin(45^\circ) = 64 \cdot \frac{\sqrt{2}}{2} = 32\sqrt{2} \approx 45.25 \text{ cm}^2 \]
Problem 14
The diagonals of a rhombus are equal and each measures 10 cm. Find the area of the rhombus.
Solution:
\[ A = \frac{d_1 \cdot d_2}{2} = \frac{10 \cdot 10}{2} = 50 \text{ cm}^2 \]
Problem 15
A rhombus has an area of 120 m⊃;2; and one diagonal measures 24 m. What is the length of the other diagonal?
Solution:
\[ A = \frac{d_1 \cdot d_2}{2} \implies 120 = \frac{24 \cdot d_2}{2} \implies d_2 = 10 \text{ m} \]
Problem 16
A rhombus has diagonals of lengths 8 cm and 6 cm. What is its area?
Solution:
\[ A = \frac{d_1 \cdot d_2}{2} = \frac{8 \cdot 6}{2} = 24 \text{ cm}^2 \]
Problem 17
A rhombus has an area of 54 m⊃;2;. If one diagonal measures 9 m, find the length of the other diagonal.
Solution:
\[ A = \frac{d_1 \cdot d_2}{2} \implies 54 = \frac{9 \cdot d_2}{2} \implies d_2 = 12 \text{ m} \]
Problem 18
A rhombus has each side measuring 6 cm and one angle measuring 120 degrees. What is the area?
Solution:
\[ A = a^2 \sin(\theta) = 6^2 \sin(120^\circ) = 36 \cdot \frac{\sqrt{3}}{2} = 18\sqrt{3} \approx 31.18 \text{ cm}^2 \]
Problem 19
The area of a rhombus is 64 cm⊃;2; and one diagonal measures 16 cm. Find the length of the other diagonal.
Solution:
\[ A = \frac{d_1 \cdot d_2}{2} \implies 64 = \frac{16 \cdot d_2}{2} \implies d_2 = 8 \text{ cm} \]
Problem 20
A rhombus has diagonals measuring 18 cm and 24 cm. What is its area?
Solution:
\[ A = \frac{d_1 \cdot d_2}{2} = \frac{18 \cdot 24}{2} = 216 \text{ cm}^2 \]
