A solid is in the form of a cone standing on a hemi-sphere with both their radii being equal to 8 cm and the height of cone is equal to its radius. Find the volume of the solid in terms of pi .

Answer

To find the volume of the solid formed by a cone standing on a hemisphere, we can break the problem down into a few steps.

### Step-by-Step Solution:

1. **Identify the dimensions**:
- The radius of both the cone and the hemisphere is given as \( r = 8 \) cm.
- The height of the cone is equal to its radius, so \( h = 8 \) cm.

2. **Calculate the volume of the cone**:
- The formula for the volume of a cone is:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
- Substituting the values:
\[ V_{\text{cone}} = \frac{1}{3} \pi (8)^2 (8) = \frac{1}{3} \pi (64)(8) = \frac{1}{3} \pi (512) = \frac{512}{3} \pi \text{ cm}^3 \]

3. **Calculate the volume of the hemisphere**:
- The formula for the volume of a hemisphere is:
\[ V_{\text{hemisphere}} = \frac{2}{3} \pi r^3 \]
- Substituting the radius:
\[ V_{\text{hemisphere}} = \frac{2}{3} \pi (8)^3 = \frac{2}{3} \pi (512) = \frac{1024}{3} \pi \text{ cm}^3 \]

4. **Calculate the total volume of the solid**:
- The total volume \( V_{\text{total}} \) is the sum of the volumes of the cone and the hemisphere:
\[ V_{\text{total}} = V_{\text{cone}} + V_{\text{hemisphere}} = \frac{512}{3} \pi + \frac{1024}{3} \pi \]
- Combining the volumes:
\[ V_{\text{total}} = \left(\frac{512 + 1024}{3}\right) \pi = \frac{1536}{3} \pi = 512 \pi \text{ cm}^3 \]

### Final Answer:
The volume of the solid in terms of \( \pi \) is:
\[ \boxed{512 \pi \text{ cm}^3} \]