To find the area of the sector QRS in circle R, we can use the formula for the area of a sector:
\[ \text{Area of Sector} = \frac{\theta}{360} \times \pi r^2 \]
where \(\theta\) is the angle in degrees and \(r\) is the radius of the circle.
From the problem, we have:
- \(m\angle QRS = 156^\circ\)
- \(QR = 10\) (which is the radius \(r\) of the circle)
Now, we can plug these values into our formula for the area of the sector:
\[ \text{Area of Sector QRS} = \frac{156}{360} \times \pi \times (10)^2 \]
Calculating \(10^2\):
\[ 10^2 = 100 \]
Now substituting this back into the formula:
\[ \text{Area of Sector QRS} = \frac{156}{360} \times \pi \times 100 \]
Calculating \(\frac{156}{360}\):
\[ \frac{156}{360} = 0.4333\ldots \approx 0.4333 \]
Now calculating the area:
\[ \text{Area of Sector QRS} = 0.4333 \times \pi \times 100 \]
\[ \text{Area of Sector QRS} = 43.33 \times \pi \]
Now, using \(\pi \approx 3.14159\):
\[ \text{Area of Sector QRS} \approx 43.33 \times 3.14159 \approx 136.79 \]
Finally, rounding to the nearest hundredth, we find:
\[ \text{Area of Sector QRS} \approx 136.79 \text{ square units} \]
So the area of sector QRS is approximately 136.79 square units.