Two sides of a parallelgram are 12 cm and 8 cm. If one of the interior angles is `135^@`, then find area of the parallelogram.

Explanation

To find the area of the parallelogram with sides of lengths 12 cm and 8 cm and an interior angle of 135 degrees, we can use the formula for the area of a parallelogram:

\[ \text{Area} = \text{base} \times \text{height} \]

However, we can also use the formula that involves the sine of the angle between the two sides:

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

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

### Step-by-Step Solution:

1. **Identify the sides and angle**:
- Let \( a = 12 \) cm (one side)
- Let \( b = 8 \) cm (the other side)
- Let \( \theta = 135^\circ \) (the angle between the sides)

2. **Calculate the sine of the angle**:
\[ \sin(135^\circ) = \sin(180^\circ - 45^\circ) = \sin(45^\circ) = \frac{1}{\sqrt{2}} \]

3. **Substitute the values into the area formula**:
\[ \text{Area} = a \times b \times \sin(\theta) = 12 \times 8 \times \sin(135^\circ) \]
\[ = 12 \times 8 \times \frac{1}{\sqrt{2}} \]

4. **Calculate the multiplication**:
\[ = 96 \times \frac{1}{\sqrt{2}} = \frac{96}{\sqrt{2}} \]

5. **Rationalize the denominator**:
\[ = \frac{96 \sqrt{2}}{2} = 48 \sqrt{2} \text{ cm}^2 \]

### Final Answer:
The area of the parallelogram is \( 48 \sqrt{2} \text{ cm}^2 \).

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