How to Determine the Length of an Arc
Knowing how to calculate the circumference of a circle and, in turn, the length of an arc — a portion of the circumference — is important in pre-calculus because you can use that information to analyze the motion of an object moving in a circle.
An arc can come from a central angle, which is one whose vertex is located at the center of the circle. You can measure an arc in two different ways:
As an angle. The measure of an arc as an angle is the same as the central angle that intercepts it.
As a length. The length of an arc is directly proportional to the circumference of the circle and is dependent on both the central angle and the radius of the circle.
If you think back to geometry, you may remember that the formula for the circumference of a circle is
with r representing the radius. Also recall that a circle has 360 degrees. So if you need to find the length of an arc, you need to figure out what part of the whole circumference (or what fraction) you’re looking at.
You use the following formula to calculate the arc length: The symbol theta (θ) represents the measure of the angle in degrees, and s represents arc length, as shown in the figure: