Which of the following correctly expresses the relationship between the length of an arc $s$ of a circle, its radius $r$, and the central angle $\theta$ in radians?

• A. $s=r\theta$
• B. $s=\frac{\theta }{r}$
• C. $s=\frac{r\theta }{2}$
• D. $s=r·{\theta }^2$

Answer: (A)

Verified step by step guidance

1. Recall the definition of the arc length \(s\) in a circle: it is the distance along the curved path of the circle subtended by a central angle \(\theta\).
2. Understand that when the central angle \(\theta\) is measured in radians, the arc length \(s\) is directly proportional to both the radius \(r\) of the circle and the angle \(\theta\) itself.
3. Use the fundamental formula for arc length in radians: \(s = r \times \theta\).
4. Note that this formula comes from the fact that the circumference of a full circle is \$2 \pi r\(, and a full circle corresponds to an angle of \)2 \pi\( radians, so the arc length is a fraction of the circumference proportional to \)\theta / (2 \pi)$.
5. Therefore, the correct relationship is \(s = r \theta\), which expresses the arc length as the product of the radius and the central angle in radians.