The perimeter of a rhombus is 40 cm and the measure of an angle is `60^(@)`, then the area of it is:

Options

A. `100 sqrt(3) cm^(2)`
B. `50sqrt(3) cm^(2)`
C. `160 sqrt(3) cm^(2)`
D. `100 cm^(2)`

Answer

To find the area of the rhombus given its perimeter and an angle, we can follow these steps:

### Step 1: Find the length of one side of the rhombus.
The perimeter of a rhombus is given by the formula:
\[ \text{Perimeter} = 4 \times \text{side} \]
Given that the perimeter is 40 cm, we can set up the equation:
\[ 40 = 4 \times \text{side} \]
Dividing both sides by 4, we find:
\[ \text{side} = \frac{40}{4} = 10 \text{ cm} \]

### Step 2: Use the formula for the area of a rhombus.
The area \( A \) of a rhombus can be calculated using the formula:
\[ A = a^2 \sin(\theta) \]
where \( a \) is the length of a side and \( \theta \) is one of the angles of the rhombus. Here, \( a = 10 \text{ cm} \) and \( \theta = 60^\circ \).

### Step 3: Calculate the area.
Now, substituting the values into the area formula:
\[ A = 10^2 \sin(60^\circ) \]
We know that \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \), so:
\[ A = 100 \times \frac{\sqrt{3}}{2} = 50\sqrt{3} \text{ cm}^2 \]

### Final Answer:
The area of the rhombus is \( 50\sqrt{3} \text{ cm}^2 \).

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