To determine the length of the arc, we use the formula:

$L = r \theta$

where:

• $L$ is the arc length,
• $r = 11 \, \text{in}$ is the radius,
• $\theta$ is the central angle in radians.

Step 1: Convert the central angle to radians

To convert degrees to radians, use the conversion factor:

$\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180}$

For $207^\circ$:

$\theta = 207 \times \frac{\pi}{180} = \frac{207\pi}{180} = \frac{23\pi}{20} \, \text{radians}.$

Approximate this value: $\theta \approx 3.613 \, \text{radians}.$

Step 2: Calculate the arc length

Substitute $r = 11$ and $\theta \approx 3.613$ into the formula:

$L = 11 \cdot 3.613 \approx 39.74 \, \text{in}.$

Final Answer:

The arc length is approximately: $\boxed{39.74 \, \text{in}}$

Would you like more details, or do you have any questions?

Here are some related questions to explore:

1. How do you find the area of a sector with the same angle?
2. What if the angle were given in radians instead of degrees?
3. How would the arc length change if the radius doubled?
4. How can you calculate the circumference of the entire circle?
5. How do we derive the formula for arc length?

Tip: Always convert the angle to radians when using the arc length formula.