The lateral surface area of a hollow cylinder of outer radius R, inner radius r and height h is_______.

Answer

To find the lateral surface area of a hollow cylinder with outer radius \( R \), inner radius \( r \), and height \( h \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Structure of a Hollow Cylinder**:
- A hollow cylinder consists of two cylindrical surfaces: one is the outer cylinder with radius \( R \) and the other is the inner cylinder with radius \( r \).

2. **Formula for Lateral Surface Area of a Cylinder**:
- The lateral surface area (curved surface area) of a solid cylinder is given by the formula:
\[ \text{Lateral Surface Area} = 2\pi \times \text{radius} \times \text{height} \]

3. **Calculate the Lateral Surface Area of the Outer Cylinder**:
- For the outer cylinder (with radius \( R \)):
\[ \text{Lateral Surface Area of Outer Cylinder} = 2\pi R h \]

4. **Calculate the Lateral Surface Area of the Inner Cylinder**:
- For the inner cylinder (with radius \( r \)):
\[ \text{Lateral Surface Area of Inner Cylinder} = 2\pi r h \]

5. **Combine the Lateral Surface Areas**:
- To find the total lateral surface area of the hollow cylinder, we add the lateral surface areas of both the outer and inner cylinders:
\[ \text{Total Lateral Surface Area} = \text{Lateral Surface Area of Outer Cylinder} + \text{Lateral Surface Area of Inner Cylinder} \]
\[ = 2\pi R h + 2\pi r h \]

6. **Factor Out Common Terms**:
- We can factor out \( 2\pi h \) from the equation:
\[ \text{Total Lateral Surface Area} = 2\pi h (R + r) \]

### Final Answer:
The lateral surface area of a hollow cylinder is:
\[ \text{Lateral Surface Area} = 2\pi h (R + r) \]

---