To find the area of sector \( FGH \) in circle \( G \), we can use the formula for the area of a sector:
\[ \text{Area of Sector} = \frac{m\angle}{360^\circ} \times \pi r^2 \]
where \( m\angle FGH \) is the measure of the angle in degrees, and \( r \) is the radius of the circle.
In this problem:
- \( m\angle FGH = 112^\circ \)
- \( FG = 14 \) (this will represent the radius, \( r \), of the circle)
Substituting these values into the formula:
\[ \text{Area of Sector} = \frac{112}{360} \times \pi \times (14)^2 \]
First, calculate \( 14^2 \):
\[ 14^2 = 196 \]
Now substitute this value back into the formula:
\[ \text{Area of Sector} = \frac{112}{360} \times \pi \times 196 \]
Next, simplify \( \frac{112}{360} \):
\[ \frac{112}{360} = \frac{14}{45} \]
Now, substitute that back into the formula:
\[ \text{Area of Sector} = \frac{14}{45} \times \pi \times 196 \]
Next, calculate \( \frac{14 \times 196}{45} \):
\[ 14 \times 196 = 2744 \]
Now, divide by 45:
\[ \frac{2744}{45} \approx 60.97777777 \]
Now, multiply by \( \pi \):
\[ \text{Area of Sector} \approx 60.97777777 \times 3.14159265 \approx 191.192 c \]
Finally, rounding this to the nearest hundredth, we have:
\[ \text{Area of Sector} \approx 191.19 \]
Thus, the area of sector \( FGH \) is approximately \( \boxed{191.19} \).