Arc Length

A circle's full turn is 360 degrees no matter how big or small the circle is. That's what degree measure tracks: a pure fraction of the whole turn. But length is a physical distance, and distance depends on the circle's radius. A 90 degree arc is always a quarter of the way around, but a quarter of the way around a basketball hoop is a lot shorter than a quarter of the way around a running track. So arc length has to bring in the radius, while degree measure never does.

The formula just says this directly: arc length equals (central angle / 360) times the circumference, where circumference is 2 π r. The angle over 360 gives you the fraction of the circle you're dealing with; multiplying that fraction by the full distance around gives you the piece of that distance. Work it in either direction: if you know the angle and radius, multiply to find length; if you know the angle and the length, set up the equation and solve for r.

This formula sits next to sector area, which uses the exact same fraction idea but multiplies by π r² instead of 2 π r. Once a central angle is in radians instead of degrees, the fraction-of-360 setup gets replaced by something cleaner: arc length equals r times the angle in radians directly, with no division step at all, since a radian is already defined in terms of arc length over radius.