State True or False:
A solid ball is exactly fitted inside the cubical box of side a. The volume of the ball is `4/3pi a^2`

Answer

To determine whether the statement "A solid ball is exactly fitted inside the cubical box of side a. The volume of the ball is \( \frac{4}{3} \pi a^2 \)" is true or false, we will analyze the situation step by step.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a cube with a side length of \( a \).
- A solid ball (sphere) is fitted inside this cube.

2. **Identifying the Dimensions**:
- The diameter of the sphere will be equal to the side length of the cube, which is \( a \).
- Therefore, the radius \( r \) of the sphere can be calculated as:
\[ r = \frac{\text{Diameter}}{2} = \frac{a}{2} \]

3. **Calculating the Volume of the Sphere**:
- The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
- Substituting the radius \( r = \frac{a}{2} \) into the volume formula:
\[ V = \frac{4}{3} \pi \left(\frac{a}{2}\right)^3 \]

4. **Simplifying the Volume Calculation**:
- First, calculate \( \left(\frac{a}{2}\right)^3 \):
\[ \left(\frac{a}{2}\right)^3 = \frac{a^3}{8} \]
- Now substitute this back into the volume formula:
\[ V = \frac{4}{3} \pi \cdot \frac{a^3}{8} \]
- Simplifying this expression:
\[ V = \frac{4 \pi a^3}{3 \cdot 8} = \frac{4 \pi a^3}{24} = \frac{\pi a^3}{6} \]

5. **Comparing with the Given Statement**:
- The statement claims that the volume of the ball is \( \frac{4}{3} \pi a^2 \).
- We found that the volume of the sphere is \( \frac{\pi a^3}{6} \).
- Since \( \frac{\pi a^3}{6} \) is not equal to \( \frac{4}{3} \pi a^2 \), the statement is false.

### Conclusion:
The statement is **False**.

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