Given a circle with radius $r$ = $6$ and an intercepted arc length of $4\pi$, what is the measure of the central angle in radians?

• A. $\frac{2\pi }{3}$
• B. $\frac{3\pi }{4}$
• C. $2\pi$
• D. $\frac{4\pi }{6}$

Answer: (D)

Verified step by step guidance

1. Recall the formula that relates the arc length \( s \), radius \( r \), and central angle \( \theta \) in radians: \[ s = r \times \theta \].
2. Identify the given values: the radius \( r = 6 \) and the arc length \( s = 4\pi \).
3. Substitute the known values into the formula: \[ 4\pi = 6 \times \theta \].
4. Solve for the central angle \( \theta \) by dividing both sides of the equation by 6: \[ \theta = \frac{4\pi}{6} \].
5. Simplify the fraction if possible to express the central angle in its simplest form.