To find the area of sector RST in circle S, we can use the following formula for the area of a sector:
\[ \text{Area of sector} = \frac{m\angle}{360^\circ} \times \pi r^2 \]
Where:
- \( m\angle \) is the measure of the angle of the sector (in degrees),
- \( r \) is the radius of the circle.
Given:
- \( m\angle RST = 120^\circ \)
- \( R S = 13 \) (which represents the radius of the circle)
Now, substituting the values into the formula:
\[ \text{Area of sector} = \frac{120^\circ}{360^\circ} \times \pi (13)^2 \]
Calculating \( \frac{120^\circ}{360^\circ} \):
\[ \frac{120}{360} = \frac{1}{3} \]
Next, calculate \( (13)^2 \):
\[ 13^2 = 169 \]
Now substituting back into the area formula:
\[ \text{Area of sector} = \frac{1}{3} \times \pi \times 169 \]
This simplifies to:
\[ \text{Area of sector} = \frac{169\pi}{3} \]
Now, let's calculate \( \frac{169\pi}{3} \):
Using \( \pi \approx 3.14159 \):
\[ \frac{169\pi}{3} \approx \frac{169 \times 3.14159}{3} \approx \frac{530.14101}{3} \approx 176.71367 \]
Rounding to the nearest hundredth:
\[ \text{Area of sector} \approx 176.71 \]
Thus, the area of sector RST is approximately 176.71 square units.