The central angle is the angle formed by drawing the radii line segments we created in the last step, and this angle measurement is the final value needed to find arc length.
If your textbook or instructor did not provide a central angle, you will have to draw two radii at the edges of the arc and then use a protractor to measure the angle where they meet at the center point.
You can take this measurement in either degrees or radians. If the central angle is greater than 180 degrees (π radians), you may want to measure the opposite side of the angle and then express it as your measurement plus 180 degrees.
Degrees vs. Radians
While degrees and radians are units used to measure the same thing, they serve slightly different purposes. One degree represents 1/360 of a circle, while a radian is a function of the ratio between radius and circumference. For this reason, a radian is equal to 1/2π.
A full circle is always 2π radians because, as you may recall, circumference is equal to 2π times the radius of the circle. We also have two handy formulas that can be used if you ever need to convert degrees to radians, or vice versa.
Degree value × (π/180) = Radian value
Radian value × (180/π) = Degree value
Fractions of a Circle
Now that we know that central angle can be used to express a fraction of a circle, and that radius has a close relationship with arc length, we have just about everything we need to calculate the arc length.
To better visualize fractions of a circle, think of a 90 degree (1/2 π radians) angle as a quarter circle, 180 degrees (π radians) as a half circle, and 360 degrees (2 π radians) as a full circles. Of course, real world examples will often be fractional values in between these neat angles.
Remembering the Arc Length Formula
Once you’ve found the central angle, you can plug it into the circumference formula from earlier to create and arc length formula. To obtain the arc length formula for degrees, take your central angle value, divide it by 360 and then multiply the result by our circumference formula.
Arc Length = (Degree Angle/360) x 2π x Arc Radius
If you calculate the arc length using radians, the process is a bit simpler. Since a radian value is a fraction of 2π, and we need to multiply by 2π in order to calculate the arc length, the two cancel out. We are left with this formula:
Arc Length = Radian Angle x Arc Radius
