A sphere and a hemisphere have the same radius. Then the ratio of their respective total surface areas is

Options

A. `2:1`
B. `1:2`
C. `4:3`
D. `3:4`

Answer

To find the ratio of the total surface areas of a sphere and a hemisphere with the same radius, we can follow these steps:

### Step 1: Understand the formulas for total surface area
- The total surface area (TSA) of a sphere is given by the formula:
\[ \text{TSA of Sphere} = 4\pi r^2 \]
- The total surface area of a hemisphere (including its base) is given by the formula:
\[ \text{TSA of Hemisphere} = 3\pi r^2 \]

### Step 2: Substitute the radius
Since both the sphere and the hemisphere have the same radius \( r \), we can directly use the formulas without any substitutions.

### Step 3: Calculate the ratio of the total surface areas
Now, we can find the ratio of the total surface areas of the sphere to the hemisphere:
\[ \text{Ratio} = \frac{\text{TSA of Sphere}}{\text{TSA of Hemisphere}} = \frac{4\pi r^2}{3\pi r^2} \]

### Step 4: Simplify the ratio
The \( \pi r^2 \) terms in the numerator and denominator cancel out:
\[ \text{Ratio} = \frac{4}{3} \]

### Step 5: State the final answer
Thus, the ratio of the total surface areas of the sphere to the hemisphere is:
\[ \text{Ratio} = 4 : 3 \]