Find the ratio of the total surface area of a sphere and a hemisphere of same radius.

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: Understand the formulas for surface areas
- 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 is given by the formula:
\[ \text{TSA of Hemisphere} = 3\pi R^2 \]
(This includes the curved surface area plus the area of the circular base.)

### Step 2: Write down the surface areas
- For a sphere with radius \( R \):
\[ \text{TSA of Sphere} = 4\pi R^2 \]
- For a hemisphere with the same radius \( R \):
\[ \text{TSA of Hemisphere} = 3\pi R^2 \]

### Step 3: Set up the ratio
- We need to find the ratio of the total surface area of the sphere to that of 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: Write the final answer
- Therefore, the ratio of the total surface area of a sphere to that of a hemisphere is:
\[ 4 : 3 \]

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