The radius of hemisphere is "r" what is its total surface area

Options

A. `2/3pir^(3)`
B. `3pir^(2)`
C. `2pir^(2)`
D. `4/3pir^(2)`

Answer

To find the total surface area of a hemisphere with radius "r", we can follow these steps:

### Step 1: Understand the Hemisphere
A hemisphere is half of a sphere. It has two surfaces: the curved surface and the flat circular base.

### Step 2: Calculate the Area of the Flat Circular Surface
The area of the flat circular surface (base) of the hemisphere can be calculated using the formula for the area of a circle:
\[ \text{Area of the flat surface} = \pi r^2 \]

### Step 3: Calculate the Curved Surface Area of the Hemisphere
The curved surface area of a hemisphere is given by the formula:
\[ \text{Curved Surface Area} = 2\pi r^2 \]

### Step 4: Add Both Areas to Find Total Surface Area
Now, to find the total surface area of the hemisphere, we need to add the area of the flat surface and the curved surface area:
\[ \text{Total Surface Area} = \text{Area of flat surface} + \text{Curved Surface Area} \]
Substituting the values we calculated:
\[ \text{Total Surface Area} = \pi r^2 + 2\pi r^2 \]

### Step 5: Simplify the Expression
Combining the terms:
\[ \text{Total Surface Area} = 3\pi r^2 \]

### Final Answer
Thus, the total surface area of the hemisphere is:
\[ \text{Total Surface Area} = 3\pi r^2 \text{ square units} \]

---