Algebra 2 · Unit 10 · Topic 10.1
Degrees, Radians, and Arc Length
A radian measures angle as arc length divided by circle radius.
Definition and explanation
A radian measures angle as arc length divided by circle radius. A full turn is 2π radians or 360 degrees, and arc length is s = rθ when θ is in radians and r and s use the same length units.
Worked example
Convert 120° to radians: 120π/180 = 2π/3. On a circle of radius 9 cm, that angle subtends arc length s = 9(2π/3) = 6π cm, approximately 18.85 cm.
Common mistake
The formula s = rθ requires radians, not a raw degree measure.
Key takeaway
A radian measures angle as arc length divided by circle radius.
Explain each step of the worked example, then check the units, signs, and conditions that apply.
Further reading
Topic coverage was checked against the openly available OpenStax Intermediate Algebra 2e, Algebra and Trigonometry 2e, and Introductory Statistics 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.