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: \(\text{Sector Area} = \frac{\theta \cdot r^{2}}{2}\).