Definition: Arc length is the distance along a circle between two points, equal to the fraction (angle / 360) times the circumference 2 pi r.
How is arc length different from arc measure?
Arc measure is an angle in degrees. Arc length is a distance in units such as centimeters. Two arcs with the same measure have different lengths on circles of different radii.
What is the formula?
Arc length = (θ / 360) · 2πr, with θ in degrees. In radians it is simply s = rθ.
Example. A circle has radius 9 cm and a central angle of 140 degrees. The arc length is (140/360)(2π)(9) = (7/18)(18π) = 7π, about 21.99 cm. Going backward, an arc of 10 cm on a circle of radius 12 cm has central angle (10 / (2π · 12)) · 360, about 47.7 degrees.
What mistakes are frequent?
Using the area formula, using the diameter where the radius is required, or leaving the answer in degrees. Another error is to forget that the whole circumference corresponds to 360 degrees. If the answer is longer than the circumference, you have made a slip.