Quick answer: An arc with central angle θ in degrees and radius r has length L = (θ/360°)2πr. In radians the formula is L = rθ; do not mix the two angle units.
Worked example
Find the arc length of a 120° sector with radius 9 cm.
- 120/360 = 1/3 of a full turn.
- The full circumference is 2π×9 = 18π cm.
- The required arc length is (1/3)18π = 6π cm.
Check the result
The arc covers a third of the circumference, as expected.
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Common mistake to avoid
Arc length excludes the two radii. The perimeter of this sector is 6π + 18 cm. Area would have square units; arc length has length units.
Try it yourself
1. Find a 90° arc for radius 4 cm. 2. Find arc length if θ = 2 radians and r = 3 cm.
Answers and reasoning
1. (1/4)2π4 = 2π cm. 2. rθ = 6 cm.
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