The ratio of the total surface area of a sphere and a hemisphere of same radius is

Options

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

Answer

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

### Step 1: Write down the formula for the total surface area of a sphere.
The total surface area \( A \) of a sphere with radius \( r \) is given by the formula:
\[ A_{\text{sphere}} = 4\pi r^2 \]

### Step 2: Write down the formula for the total surface area of a hemisphere.
The total surface area \( A \) of a hemisphere with radius \( r \) is given by the formula:
\[ A_{\text{hemisphere}} = 3\pi r^2 \]

### Step 3: Set up the ratio of the total surface area of the sphere to that of the hemisphere.
Now, we can find the ratio of the total surface area of the sphere to that of the hemisphere:
\[ \text{Ratio} = \frac{A_{\text{sphere}}}{A_{\text{hemisphere}}} = \frac{4\pi r^2}{3\pi r^2} \]

### Step 4: Simplify the ratio.
In this expression, \( \pi r^2 \) cancels out:
\[ \text{Ratio} = \frac{4}{3} \]

### Step 5: Write the ratio in a standard form.
Thus, the ratio of the total surface area of a sphere to that of a hemisphere is:
\[ 4 : 3 \]

### Conclusion:
Therefore, the ratio of the total surface area of a sphere and a hemisphere of the same radius is \( 4 : 3 \).

---