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Arc Length Examples (Degrees & Radians)

Worked arc length examples in degrees and radians for geometry homework, engineering checks, and classroom practice with full calculations.

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Blueprint-style circular sector diagram for arc length guides

Quick Answer

Substitute θ and r into L = (θ/360°) × 2πr for degrees, or L = rθ after converting to radians.

Formula

  • Degrees: L = (θ/360°) × 2πr
  • Radians: L = rθ
  • Check: L ≤ 2πr

Introduction

Worked examples turn formulas into muscle memory. Each problem below shows the given data, the chosen formula, and the arithmetic path to arc length.

Verify your own work with the Arc Length Calculator after trying each problem on paper.

If any symbol is unfamiliar, review the arc length formula article before continuing.

Main Content

How to study with these examples

Cover the final line of each problem and attempt the multiplication yourself before reading the answer.

Repeat each problem using the alternate unit system: convert Example 1 to radians and confirm you still reach 2π meters.

When a problem starts from diameter, read arc length from diameter for the same conversion logic applied to different numbers.

Worked problems

Example 1 (degrees): θ = 120°, r = 3 m. L = (120/360) × 2π(3) = (1/3) × 6π = 2π ≈ 6.283 m.

Example 2 (radians): θ = π/4 rad, r = 8 m. L = 8 × (π/4) = 2π ≈ 6.283 m.

Example 3 (quarter circle from diameter): d = 10 m, θ = 90°. Radius is 5 m. L = (90/360) × 2π(5) = 0.25 × 10π ≈ 7.854 m.

Example 4 (engineering context): A conveyor bend with r = 2.5 m and θ = 45° gives L = (45/360) × 2π(2.5) ≈ 1.963 m of belt contact along the curve.

Example 5 (comparison): For θ = 60° and r = 6 m, arc length is ≈ 6.283 m while chord length is 6 m. The difference matters when ordering curved trim rather than straight stock.

FAQ

Should I round π?
Use 3.14159 or your calculator value for decimals. Leave π symbolic when the problem requests an exact answer.
Why do Examples 1 and 2 both give 2π?
120° equals π/3 rad only when paired with different radii in other problems. Here the numbers were chosen to show equivalent methods, not identical sectors.

Conclusion

Pattern recognition speeds exams and field checks: identify unit type, write the formula, substitute, simplify.

Build a habit of unit and reasonableness checks after every calculation.

Check your answers