Given a circle with circumference $C$ and a central angle of $\theta$ radians, what is the length of the arc subtended by the angle $\theta$?

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

Answer: (B)

Verified step by step guidance

1. Recall the formula for the circumference of a circle: \(C = 2 \pi r\), where \(r\) is the radius of the circle.
2. Understand that the length of an arc \(s\) subtended by a central angle \(\theta\) (in radians) is given by the proportion of the angle to the full angle of the circle (which is \(2\pi\) radians). This can be expressed as \(s = r \times \theta\).
3. Since we know the circumference \(C = 2 \pi r\), solve for the radius \(r\) by rearranging the formula: \(r = \frac{C}{2 \pi}\).
4. Substitute the expression for \(r\) into the arc length formula: \(s = \left( \frac{C}{2 \pi} \right) \times \theta\).
5. Simplify the expression to get the arc length in terms of \(C\) and \(\theta\): \(s = \frac{\theta \cdot C}{2 \pi}\).