If the radius of a circle is tripled, what is the ratio of the area of half the original circle to the area of the whole new circle? (Area of a circle `=pir^(2)`, where r = radius)

Answer

To solve the problem, we need to find the ratio of the area of half the original circle to the area of the whole new circle after the radius has been tripled.

### Step-by-Step Solution:

1. **Identify the original radius and area:**
Let the original radius of the circle be \( r \).
The area of the original circle is given by the formula:
\[ \text{Area}_{\text{original}} = \pi r^2 \]

2. **Calculate the area of half the original circle:**
The area of half the original circle is:
\[ \text{Area}_{\text{half original}} = \frac{1}{2} \times \text{Area}_{\text{original}} = \frac{1}{2} \times \pi r^2 = \frac{\pi r^2}{2} \]

3. **Determine the new radius after tripling:**
If the radius is tripled, the new radius becomes:
\[ \text{New radius} = 3r \]

4. **Calculate the area of the new circle:**
The area of the new circle is:
\[ \text{Area}_{\text{new}} = \pi (3r)^2 = \pi \times 9r^2 = 9\pi r^2 \]

5. **Set up the ratio of the area of half the original circle to the area of the whole new circle:**
We need to find the ratio:
\[ \text{Ratio} = \frac{\text{Area}_{\text{half original}}}{\text{Area}_{\text{new}}} = \frac{\frac{\pi r^2}{2}}{9\pi r^2} \]

6. **Simplify the ratio:**
\[ \text{Ratio} = \frac{\frac{\pi r^2}{2}}{9\pi r^2} = \frac{1}{2} \times \frac{1}{9} = \frac{1}{18} \]

### Final Answer:
The ratio of the area of half the original circle to the area of the whole new circle is:
\[ \frac{1}{18} \]