The image shows a circle with a central angle of $120^\circ$ and a minor arc $\overset{\frown}{AB}$ of length 15 units. We are asked to find the circumference of the circle.

Step-by-step solution:

1. Relation between arc length and circumference: The length of an arc $L$ is given by the formula: $L = \frac{\theta}{360^\circ} \times C$ where:

   • $L$ is the arc length (in this case, 15 units),
   • $\theta$ is the central angle (120° here),
   • $C$ is the circumference of the circle.

2. Substitute the given values: We have: $15 = \frac{120^\circ}{360^\circ} \times C$

3. Simplify the equation: $15 = \frac{1}{3} \times C$ Multiply both sides by 3 to solve for $C$: $C = 45$

Final Answer:

The circumference of the circle is 45 units.

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

Here are 5 related questions to expand your understanding:

1. How do you calculate the radius of the circle if you know the circumference?
2. What is the formula for the area of a sector, and how would it apply here?
3. How can you calculate the area of the entire circle using the circumference?
4. What is the length of the major arc $\overset{\frown}{AB}$ in this circle?
5. How would the arc length formula change if the angle was given in radians?

Tip: For problems involving arcs, always remember that the arc length is proportional to the central angle.