Which of the following formulas could be used to calculate the area of a sector in a circle with radius $r$ and central angle $\theta$ (in radians)?

• A. $\frac{\theta }{2}{r}^2$
• B. $r^2\times \theta$
• C. $\frac{1}{2}{r}^2\times \sin \left(\theta \right)$
• D. $\frac{\theta }{360}\times r^2\times \pi$

Answer: (A)

Verified step by step guidance

1. Recall that the area of a sector of a circle is a fraction of the area of the entire circle, where the fraction is determined by the central angle \( \theta \) compared to the full angle of the circle.
2. Since the angle \( \theta \) is given in radians, the full angle of the circle is \( 2\pi \) radians, so the fraction of the circle represented by the sector is \( \frac{\theta}{2\pi} \).
3. The area of the entire circle is given by the formula \( \pi r^2 \), where \( r \) is the radius of the circle.
4. To find the area of the sector, multiply the fraction of the circle by the total area: \( \text{Area of sector} = \frac{\theta}{2\pi} \times \pi r^2 \).
5. Simplify the expression by canceling \( \pi \) in numerator and denominator, resulting in \( \text{Area of sector} = \frac{1}{2} r^2 \theta \), which matches the formula \( \frac{\theta}{2} r^2 \).