Area of a circular sector Prove that the area of a sector of a circle of radius r associated with a central angle $\theta$ (measured in radians) is $A=\frac12r^2\theta$.
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Verified step by step guidance

1. Start by understanding the relationship between the area of a circle and the area of a sector. The area of a full circle with radius r is given by the formula A = \pi r^2.
2. Recognize that a sector is a portion of the circle, defined by a central angle \( \theta \) (in radians). The fraction of the circle's area that the sector occupies is \( \frac{\theta}{2\pi} \), since the full circle corresponds to an angle of \( 2\pi \) radians.
3. Calculate the area of the sector by multiplying the fraction of the circle's area by the total area of the circle: \( A_{\text{sector}} = \frac{\theta}{2\pi} \times \pi r^2 \).
4. Simplify the expression: \( A_{\text{sector}} = \frac{\theta}{2\pi} \times \pi r^2 = \frac{\theta r^2}{2} \).
5. Conclude that the area of the sector is \( A = \frac{1}{2} r^2 \theta \), which is the desired formula.