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Arc Length and Sector Area (L = 2A/r)

Connect sector area and arc length with A = ½r²θ and the shortcut L = 2A/r. Find curved distance when area and radius are known.

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Blueprint-style circular sector diagram for arc length guides

Quick Answer

A = ½r²θ and L = rθ (θ in radians) combine to give L = 2A/r when area and radius are known.

Formula

  • A = ½ r² θ
  • L = r θ
  • L = 2A / r
  • θ must be radians in area formula

Introduction

Sector area and arc length describe the same pie-slice region from different angles. Both depend on radius and central angle. If one pair is known, you can often find the other without drawing extra lines.

Shaded regions on plans frequently show area. Fabrication and trimming still need the curved boundary length. Linking the two formulas saves a conversion step.

The calculator accepts sector area with radius or angle and returns arc length directly.

Main Content

Why area and arc move together

Increasing the central angle adds area and lengthens the arc at the same time. Shrinking the radius reduces both quantities for a fixed angle.

The shortcut L = 2A/r avoids solving for θ explicitly. That is valuable when a CAD export lists area and radius but omits the angle dimension.

Area formulas require radians. If your angle is in degrees, convert before using A = ½r²θ, or stay in degree arc formulas from the arc length formula article instead.

When the diagram shows diameter rather than radius, convert first using the steps in arc length from diameter, then apply L = 2A/r with the corrected radius.

Linked formulas

  • A = ½ r² θ (θ radians)
  • L = r θ
  • Eliminate θ → L = 2A/r

From area to arc

Given A = 15.708 m² and r = 5 m. L = 2(15.708)/5 ≈ 6.283 m.

Check: 6.283 m is πr for r = 5, which matches a 90° sector (quarter circle). Area A = ½(25)(π/2) ≈ 19.635 m² for that angle; numbers must come from the same sector.

Always confirm that area and radius describe the same slice before applying L = 2A/r.

FAQ

Does L = 2A/r work in any units?
Yes, provided area uses the square of the linear unit for radius and arc.
Can I use chord instead of arc with area?
Area is tied to central angle and radius, not chord. Convert through angle if chord is the only extra measurement.

Conclusion

Sector area and arc length share θ and r. When angle is hidden, L = 2A/r is the fast bridge.

Label the sector clearly, keep radians inside area formulas, and verify with the calculator when possible.

Calculate from area