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Arc Length vs Chord Length: Key Differences

Compare arc length and chord length with formulas, diagrams, and common mistakes. Learn when to measure the curve versus the straight line.

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Blueprint-style circular sector diagram for arc length guides

Quick Answer

Arc: L = rθ (θ in radians). Chord: c = 2r sin(θ/2). For any nonzero angle, arc length exceeds chord length.

Formula

  • Arc along circumference
  • Chord straight across
  • L > c when θ > 0
  • Same endpoints, different paths

Introduction

Arc length and chord length describe two different paths between the same pair of points on a circle. Arc follows the curve; chord cuts straight across.

Mixing them on a blueprint or exam diagram is a common source of wrong orders and lost exam marks. Label each measurement before you calculate.

The home calculator accepts chord length paired with radius or angle and returns arc length, which is usually the value needed for curved material.

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Geometric interpretation

Picture a circular gate hinge. The hinge leaf travels along an arc as the gate opens. A string pulled tight between the hinge endpoints measures chord length. The gate hardware must follow the arc, not the string.

As the central angle increases toward 180°, chord length approaches the diameter while arc length approaches half the circumference. The gap between the two values is largest near 180° for a given radius.

Students sometimes compute chord when the word arc appears in the prompt. Re-read the question and trace the curved path with your finger on the diagram.

The definition article on what is arc length contrasts the two paths with everyday language before formulas appear.

Once you know chord and radius, you can recover the central angle and then arc length. The relationship is developed further in arc length and central angle.

Side-by-side formulas

  • Arc: L = rθ (θ radians)
  • Chord: c = 2r sin(θ/2)
  • Degree arc: L = (θ/360°) × 2πr

Numeric comparison

θ = 90°, r = 5 m. Arc L = (90/360) × 2π(5) ≈ 7.854 m. Chord c = 2(5) sin(45°) ≈ 7.071 m.

θ = 60°, r = 5 m. Arc ≈ 5.236 m. Chord = 5 m exactly. Arc still wins, but by a smaller margin.

FAQ

Can chord equal arc?
Only in the limit as the angle approaches zero. For any visible sector, arc is longer.
Which value do fabricators need?
Usually arc length, because material bends along the curve.

Conclusion

Arc and chord answer different measurement questions. Identify the path first, then choose the formula.

When chord is known, the calculator can still return arc length if you supply a second measurement.

Convert chord to arc