Find the area of `Delta ` whose length of medians is 18 cm 24 cm & 30 cm.

Answer

To find the area of a triangle given the lengths of its medians, we can use the formula:

\[ \text{Area} = \frac{4}{3} \sqrt{s \cdot (s - m_1) \cdot (s - m_2) \cdot (s - m_3)} \]

where \( m_1, m_2, m_3 \) are the lengths of the medians, and \( s \) is the semi-perimeter of the triangle formed by the medians, calculated as:

\[ s = \frac{m_1 + m_2 + m_3}{2} \]

### Step-by-Step Solution:

1. **Identify the lengths of the medians**:
- Let \( m_1 = 18 \, \text{cm} \)
- Let \( m_2 = 24 \, \text{cm} \)
- Let \( m_3 = 30 \, \text{cm} \)

2. **Calculate the semi-perimeter \( s \)**:
\[ s = \frac{m_1 + m_2 + m_3}{2} = \frac{18 + 24 + 30}{2} = \frac{72}{2} = 36 \, \text{cm} \]

3. **Substitute values into the area formula**:
\[ \text{Area} = \frac{4}{3} \sqrt{s \cdot (s - m_1) \cdot (s - m_2) \cdot (s - m_3)} \]

4. **Calculate \( s - m_1, s - m_2, s - m_3 \)**:
- \( s - m_1 = 36 - 18 = 18 \)
- \( s - m_2 = 36 - 24 = 12 \)
- \( s - m_3 = 36 - 30 = 6 \)

5. **Substitute these values back into the area formula**:
\[ \text{Area} = \frac{4}{3} \sqrt{36 \cdot 18 \cdot 12 \cdot 6} \]

6. **Calculate the product under the square root**:
- First, calculate \( 36 \cdot 18 \):
\[ 36 \cdot 18 = 648 \]
- Next, calculate \( 12 \cdot 6 \):
\[ 12 \cdot 6 = 72 \]
- Now, multiply these results:
\[ 648 \cdot 72 = 46656 \]

7. **Calculate the square root**:
\[ \sqrt{46656} = 216 \]

8. **Final calculation of the area**:
\[ \text{Area} = \frac{4}{3} \cdot 216 = 288 \, \text{cm}^2 \]

### Final Answer:
The area of the triangle is \( 288 \, \text{cm}^2 \).