The formula for calculating the arc length of a circle is given by:

To gain a better understanding of this formula, let’s break it down step by step. The arc length refers to the distance along the curved line that constitutes the arc. In a circle, the total circumference, which is the distance around the entire circle, is expressed by the formula $2\pi r$, where $r$ represents the radius of the circle. However, an arc only covers a portion of this circumference, not the entire length.

In the formula, the angle $\theta$ (theta) denotes the central angle of the arc and is measured in degrees. Since a complete circle encompasses $360^\circ$, the fraction $\frac{\theta}{360^\circ}$ indicates the proportion of the full circle’s circumference that the arc represents. By multiplying this fraction by the total circumference ($2\pi r$), we can derive the length of the arc.

For instance, consider a circle with a radius of $5 \, \text{cm}$ and an arc that subtends a central angle of $60^\circ$. To find the length of this arc, you would substitute the values into the formula:

Simplifying this expression gives:

This formula is particularly useful in various practical applications, such as calculating the distance a wheel travels along a curved path or determining the length of a segment of a circular track. Remember, the essential components in this calculation are the radius of the circle and the central angle of the arc.