Given a circle with radius $r$ and a central angle $\theta$ measured in radians, what is the area of the shaded sector formed by this angle?

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

Answer: (A)

Verified step by step guidance

1. Recall that the area of a full circle is given by the formula \(\text{Area} = \pi \times r^{2}\), where \(r\) is the radius of the circle.
2. Understand that the central angle \(\theta\) (in radians) represents a fraction of the full circle. Since a full circle corresponds to an angle of \(2\pi\) radians, the fraction of the circle covered by the sector is \(\frac{\theta}{2\pi}\).
3. To find the area of the sector, multiply the total area of the circle by the fraction of the circle represented by the angle \(\theta\). This gives \(\text{Sector Area} = \left( \frac{\theta}{2\pi} \right) \times \pi r^{2}\).
4. Simplify the expression by canceling \(\pi\) in numerator and denominator, resulting in \(\text{Sector Area} = \frac{\theta \times r^{2}}{2}\).
5. Thus, the formula for the area of a sector with radius \(r\) and central angle \(\theta\) (in radians) is \(\boxed{\frac{\theta \times r^{2}}{2}}\).