The adjacent sides of a parallelogram are 6 cm and 8 cm and the angle between them is `30^(@)` . What is the area of the parallelogram ?

Options

A. `24 cm^(2) `
B. `12 cm^(2) `
C. `40 cm^(2) `
D. `24sqrt(3) cm^(2) `

Answer

To find the area of the parallelogram with adjacent sides of lengths 6 cm and 8 cm, and an angle of 30 degrees between them, we can use the formula for the area of a parallelogram:

\[ \text{Area} = \text{base} \times \text{height} = a \times b \times \sin(\theta) \]

where:
- \( a \) and \( b \) are the lengths of the adjacent sides,
- \( \theta \) is the angle between the sides.

### Step 1: Identify the values
Here, we have:
- \( a = 6 \) cm (one side)
- \( b = 8 \) cm (the adjacent side)
- \( \theta = 30^\circ \)

### Step 2: Substitute the values into the formula
Now, substitute the values into the area formula:

\[ \text{Area} = 6 \times 8 \times \sin(30^\circ) \]

### Step 3: Calculate \(\sin(30^\circ)\)
We know that:

\[ \sin(30^\circ) = \frac{1}{2} \]

### Step 4: Substitute \(\sin(30^\circ)\) into the equation
Now substitute \(\sin(30^\circ)\) back into the area formula:

\[ \text{Area} = 6 \times 8 \times \frac{1}{2} \]

### Step 5: Simplify the expression
Now simplify the expression:

\[ \text{Area} = 6 \times 8 \times 0.5 = 6 \times 4 = 24 \text{ cm}^2 \]

### Final Answer
Thus, the area of the parallelogram is:

\[ \text{Area} = 24 \text{ cm}^2 \]

---