Given that the measure of central angle $\mathrm{QRS}$ is $\theta$ radians in a circle of radius $r$, what is the area of the shaded sector? Choose the correct formula from the options below.

• A. $\frac{\theta }{2}$
• B. $\frac{\theta \cdot r^2}{2}$
• C. $\frac{\theta \cdot r}{2}$
• D. $\frac{r^2}{\theta }$

Answer: (B)

Verified step by step guidance

1. Recall that the area of a sector of a circle is proportional to the measure of its central angle. The full circle has an angle of \$2\pi\( radians and an area of \)\pi r^{2}$.
2. To find the area of the sector with central angle \(\theta\), set up a ratio comparing the sector's angle to the full circle's angle: \(\frac{\theta}{2\pi}\).
3. Multiply this ratio by the total area of the circle to get the sector area: \(\text{Area of sector} = \frac{\theta}{2\pi} \times \pi r^{2}\).
4. Simplify the expression by canceling \(\pi\) in numerator and denominator: \(\text{Area of sector} = \frac{\theta r^{2}}{2}\).
5. Thus, the formula for the area of the sector with central angle \(\theta\) radians and radius \(r\) is \(\frac{\theta \cdot r^{2}}{2}\).