The area of a parallelogram is 243 sq.cm and its height is one-third of its base. Find the base.

Answer

To find the base of the parallelogram given that its area is 243 sq.cm and its height is one-third of its base, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the formula for the area of a parallelogram**:
The area \( A \) of a parallelogram is given by the formula:
\[ A = \text{base} \times \text{height} \]

2. **Set up the relationship between height and base**:
We are given that the height \( h \) is one-third of the base \( b \). This can be expressed as:
\[ h = \frac{1}{3}b \]

3. **Substitute the height in the area formula**:
Substitute \( h \) in the area formula:
\[ A = b \times h = b \times \left(\frac{1}{3}b\right) \]
This simplifies to:
\[ A = \frac{1}{3}b^2 \]

4. **Set the area equal to the given area**:
We know the area is 243 sq.cm, so we can set up the equation:
\[ \frac{1}{3}b^2 = 243 \]

5. **Multiply both sides by 3 to eliminate the fraction**:
\[ b^2 = 243 \times 3 \]
\[ b^2 = 729 \]

6. **Take the square root of both sides to find the base**:
\[ b = \sqrt{729} \]
\[ b = 27 \]

### Final Answer:
The base of the parallelogram is \( 27 \) cm.