To determine the angle at the center of a circle, you can use the following formula:

The angle at the center of a circle is defined as the angle formed by two radii that extend from the center to the circumference. To calculate this angle, you must know two key measurements: the arc length (the distance along the curved portion of the circle) and the circumference (the total distance around the circle).

To find the angle, follow these steps:

1. Measure the Arc Length: This is the length of the curved edge of the circle between the two radii.
2. Calculate the Circumference: Use the formula: $$\text{circumference} = 2\pi r,$$ where $r$ is the radius of the circle.
3. Calculate the Fraction: Divide the arc length by the circumference to determine the fraction of the circle that the arc represents: $$\text{fraction} = \frac{\text{arc length}}{\text{circumference}}.$$
4. Convert to Angle: Multiply this fraction by $360^\circ$ to obtain the central angle.

For example, if the arc length is $5 \, \text{cm}$ and the circumference is $20 \, \text{cm}$, the fraction of the circle is:

Multiplying this fraction by $360^\circ$ yields a central angle of:

This method allows you to accurately determine the central angle using the arc length and the total circumference of the circle.