The ratio between the curved surface area and the total surface area of a cyclinder is 1: 2 Find the ratio between the height and the radius of the cyclinder.

Answer

To solve the problem of finding the ratio between the height and the radius of a cylinder given that the ratio between the curved surface area and the total surface area is 1:2, we will follow these steps:

### Step-by-Step Solution:

1. **Understand the Formulas**:
- The **Curved Surface Area (CSA)** of a cylinder is given by the formula:
\[ \text{CSA} = 2\pi rh \]
- The **Total Surface Area (TSA)** of a cylinder is given by the formula:
\[ \text{TSA} = 2\pi r(r + h) \]

2. **Set Up the Ratio**:
- According to the problem, the ratio of the curved surface area to the total surface area is:
\[ \frac{\text{CSA}}{\text{TSA}} = \frac{1}{2} \]

3. **Substitute the Formulas into the Ratio**:
- Substitute the formulas for CSA and TSA into the ratio:
\[ \frac{2\pi rh}{2\pi r(r + h)} = \frac{1}{2} \]

4. **Simplify the Equation**:
- Cancel \(2\pi r\) from both the numerator and the denominator:
\[ \frac{h}{r + h} = \frac{1}{2} \]

5. **Cross-Multiply to Solve for h**:
- Cross-multiply to eliminate the fraction:
\[ 2h = r + h \]

6. **Rearrange the Equation**:
- Rearranging the equation gives:
\[ 2h - h = r \]
\[ h = r \]

7. **Find the Ratio of Height to Radius**:
- The ratio of height (h) to radius (r) is:
\[ \frac{h}{r} = \frac{r}{r} = 1 \]
- Therefore, the ratio of height to radius is:
\[ 1:1 \]

### Final Answer:
The ratio between the height and the radius of the cylinder is \(1:1\).