Ratio of total surface area of hemisphere to that of sphere ?

Options

A. `27:8`
B. `21:4`
C. `27:4`
D. `3:4`

Answer

To find the ratio of the total surface area of a hemisphere to that of a sphere, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formulas**:
- The total surface area of a hemisphere is given by the formula:
\[ \text{Total Surface Area of Hemisphere} = 3\pi r^2 \]
- The total surface area of a sphere is given by the formula:
\[ \text{Total Surface Area of Sphere} = 4\pi r^2 \]

2. **Set Up the Ratio**:
- We need to find the ratio of the total surface area of the hemisphere to that of the sphere:
\[ \text{Required Ratio} = \frac{\text{Total Surface Area of Hemisphere}}{\text{Total Surface Area of Sphere}} = \frac{3\pi r^2}{4\pi r^2} \]

3. **Simplify the Ratio**:
- In the fraction, the terms \( \pi r^2 \) in the numerator and denominator cancel out:
\[ \frac{3\pi r^2}{4\pi r^2} = \frac{3}{4} \]

4. **Express the Ratio in Standard Form**:
- The ratio \( \frac{3}{4} \) can be expressed as:
\[ 3 : 4 \]

5. **Final Answer**:
- Therefore, the ratio of the total surface area of a hemisphere to that of a sphere is:
\[ 3 : 4 \]