The internal and external diameters of a hollow hemispherical vessel are 21 cm and 28 cm respectively. Find :
total surface area,

Answer

To find the total surface area of a hollow hemispherical vessel with given internal and external diameters, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Internal diameter (d) = 21 cm
- External diameter (D) = 28 cm

2. **Calculate the Internal and External Radii:**
- Internal radius (r) = d/2 = 21 cm / 2 = 10.5 cm
- External radius (R) = D/2 = 28 cm / 2 = 14 cm

3. **Understand the Components of the Total Surface Area:**
- The total surface area of the hollow hemispherical vessel consists of:
- The external curved surface area of the outer hemisphere
- The internal curved surface area of the inner hemisphere
- The area of the circular disc at the base

4. **Calculate the External Curved Surface Area:**
- The formula for the curved surface area of a hemisphere is given by:
\[ \text{Curved Surface Area} = 2\pi R^2 \]
- Substituting the external radius:
\[ \text{External Curved Surface Area} = 2\pi(14)^2 = 2\pi(196) = 392\pi \text{ cm}^2 \]

5. **Calculate the Internal Curved Surface Area:**
- Using the same formula for the internal radius:
\[ \text{Internal Curved Surface Area} = 2\pi r^2 \]
- Substituting the internal radius:
\[ \text{Internal Curved Surface Area} = 2\pi(10.5)^2 = 2\pi(110.25) = 220.5\pi \text{ cm}^2 \]

6. **Calculate the Area of the Circular Disc:**
- The area of the disc is given by:
\[ \text{Area of Disc} = \pi(R^2 - r^2) \]
- Substituting the values:
\[ \text{Area of Disc} = \pi(14^2 - 10.5^2) = \pi(196 - 110.25) = \pi(85.75) = 85.75\pi \text{ cm}^2 \]

7. **Combine All Areas to Find the Total Surface Area:**
- Total Surface Area = External Curved Surface Area + Internal Curved Surface Area + Area of Disc
- Therefore:
\[ \text{Total Surface Area} = 392\pi + 220.5\pi + 85.75\pi = (392 + 220.5 + 85.75)\pi = 698.25\pi \text{ cm}^2 \]

8. **Substitute the Value of \(\pi\) (Using \(\frac{22}{7}\)):**
- Total Surface Area = \(698.25 \times \frac{22}{7}\)
- Calculating this gives:
\[ \text{Total Surface Area} \approx 698.25 \times 3.14 \approx 2194.6 \text{ cm}^2 \]

### Final Answer:
The total surface area of the hollow hemispherical vessel is approximately **2194.6 cm²**.