The total surface area of a cylinder of base radius r and height h is______.

Answer

To find the total surface area of a cylinder with base radius \( r \) and height \( h \), we need to consider three components: the area of the two circular bases and the lateral (curved) surface area.

### Step-by-Step Solution:

1. **Identify the Components of the Surface Area**:
- A cylinder has two circular bases (top and bottom) and a curved surface area around the sides.

2. **Calculate the Area of the Circular Bases**:
- The area of one circular base is given by the formula:
\[ \text{Area of base} = \pi r^2 \]
- Since there are two bases, the total area of the bases is:
\[ \text{Total area of bases} = 2 \times \pi r^2 = 2\pi r^2 \]

3. **Calculate the Curved Surface Area**:
- The curved surface area (lateral surface area) of the cylinder can be calculated using the formula:
\[ \text{Curved Surface Area} = 2\pi rh \]

4. **Combine the Areas to Find the Total Surface Area**:
- The total surface area of the cylinder is the sum of the area of the two bases and the curved surface area:
\[ \text{Total Surface Area} = \text{Area of bases} + \text{Curved Surface Area} \]
- Substituting the values we calculated:
\[ \text{Total Surface Area} = 2\pi r^2 + 2\pi rh \]

5. **Factor the Expression**:
- We can factor out \( 2\pi \) from the expression:
\[ \text{Total Surface Area} = 2\pi (r + h) \]

### Final Answer:
The total surface area of a cylinder with base radius \( r \) and height \( h \) is:
\[ \text{Total Surface Area} = 2\pi (r + h) \]