If the measure of central angle RST is $\theta$ radians in a circle of radius $r$, what is the area of the shaded sector formed by this angle?

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

Answer: (C)

Verified step by step guidance

1. Recall that the area of a sector of a circle is a fraction of the total area of the circle, where the fraction is determined by the ratio of the central angle to the full angle of the circle (which is \$2\pi$ radians).
2. The total area of a circle with radius \(r\) is given by the formula \(\pi r^{2}\).
3. Since the central angle \(\theta\) is in radians, the fraction of the circle's area that the sector occupies is \(\frac{\theta}{2\pi}\).
4. Multiply the total area of the circle by this fraction to find the area of the sector: \(\text{Area of sector} = \pi r^{2} \times \frac{\theta}{2\pi}\).
5. Simplify the expression by canceling \(\pi\) in numerator and denominator, resulting in the formula for the area of the sector: \(\frac{\theta r^{2}}{2}\).