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

Answer: (A)

Verified step by step guidance

1. Recall that the area of a full circle is given by the formula \(\text{Area} = \pi \cdot r^{2}\), where \(r\) is the radius of the circle.
2. Understand that the central angle \(\theta\) (in radians) defines a sector of the circle, which is a 'slice' of the circle proportional to the angle \(\theta\) compared to the full angle of \$2\pi$ radians.
3. Since the full circle corresponds to an angle of \$2\pi\( radians, the fraction of the circle's area represented by the sector is \)\frac{\theta}{2\pi}$.
4. Multiply the total area of the circle by this fraction to find the area of the sector: \(\text{Sector Area} = \frac{\theta}{2\pi} \times \pi r^{2}\).
5. Simplify the expression by canceling \(\pi\) in numerator and denominator to get the formula for the sector area: \(\text{Sector Area} = \frac{\theta \cdot r^{2}}{2}\).