We are given a circle with:

• Radius $r = 7 \, \text{cm}$
• Arc length $AB = 11 \, \text{cm}$

The task is to find the central angle $\angle AOB$ in degrees.

We can use the formula for arc length $s$ to relate the arc and the angle:

$s = r \theta$

where:

• $s$ is the arc length,
• $r$ is the radius,
• $\theta$ is the angle in radians.

First, solve for $\theta$:

$\theta = \frac{s}{r} = \frac{11 \, \text{cm}}{7 \, \text{cm}} = 1.5714 \, \text{radians}$

To convert radians to degrees:

$\theta_{\text{degrees}} = \theta \times \frac{180^\circ}{\pi} \approx 1.5714 \times \frac{180^\circ}{3.1416} \approx 90^\circ$

Thus, the central angle $\angle AOB$ is approximately $90^\circ$.

Let me know if you'd like more details or have any questions!

Here are 5 related questions you might want to explore:

1. How would the angle change if the arc length were longer or shorter?
2. Can you derive the formula for the circumference of a circle using this concept?
3. How would you calculate the area of the sector subtended by $\angle AOB$?
4. What would the arc length be if $\angle AOB$ were $45^\circ$?
5. How would you find the length of chord $AB$?

Tip: Remember that arc length directly depends on both the radius and the angle, so increasing either will increase the arc's length.