If curved surface area of cylinder is equal to its volume. What is the radius of cylinder?

Answer

To solve the problem where the curved surface area of a cylinder is equal to its volume, we can follow these steps:

### Step-by-Step Solution:

1. **Define Variables**:
Let the radius of the cylinder be \( R \) and the height be \( H \).

2. **Write the Formulas**:
- The formula for the curved surface area (CSA) of a cylinder is:
\[ \text{CSA} = 2\pi RH \]
- The formula for the volume (V) of a cylinder is:
\[ V = \pi R^2 H \]

3. **Set the Equations Equal**:
According to the problem, the curved surface area is equal to the volume:
\[ 2\pi RH = \pi R^2 H \]

4. **Simplify the Equation**:
We can simplify this equation by cancelling out common terms. First, divide both sides by \( \pi \) (assuming \( \pi \neq 0 \)):
\[ 2RH = R^2H \]
Next, divide both sides by \( H \) (assuming \( H \neq 0 \)):
\[ 2R = R^2 \]

5. **Rearrange the Equation**:
Rearranging gives us:
\[ R^2 - 2R = 0 \]

6. **Factor the Equation**:
Factoring out \( R \):
\[ R(R - 2) = 0 \]

7. **Solve for R**:
This gives us two solutions:
\[ R = 0 \quad \text{or} \quad R = 2 \]
Since a radius cannot be zero, we take:
\[ R = 2 \]

### Final Answer:
The radius of the cylinder is \( R = 2 \).