Arc Length & Sector Area Calculator

Frequently Asked Questions

How are arc length and sector area calculated?

With the central angle θ in radians: arc length = rθ, sector area = ½r²θ. In degrees, arc = 2πr·(θ/360) and sector = πr²·(θ/360).

What is the difference between a sector and a segment?

A sector is the "pie slice" bounded by two radii and the arc. A segment is the region between a chord and its arc - sector area minus the triangle.

How is the chord length found?

Chord = 2r·sin(θ/2) for central angle θ. The tool reports it alongside arc, sector, and segment.

Why use radians?

Arc = rθ only holds when θ is in radians (radian is defined as arc/radius). The tool converts degrees automatically.

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Estimates for informational purposes only.

Important Disclaimer: Estimates for informational purposes only.

This calculator provides estimates for informational purposes only. Results are based on assumptions and may not reflect actual outcomes. Consult qualified professionals in relevant fields before making important decisions based on these results.

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How This Calculator Works

A sector is the pie-slice region bounded by two radii and the arc between them; everything follows from the central angle θ measured in radians. Arc length is s = rθ - the definition of the radian, since one radian subtends an arc exactly equal to the radius. Sector area is A = ½r²θ, the radian analogue of ½base×height. The straight chord connecting the arc's endpoints is c = 2r·sin(θ÷2) from the isosceles triangle formed by the two radii. The segment is the region between the chord and the arc; its area is the sector minus the triangle: ½r²θ − ½r²sinθ = ½r²(θ − sinθ). Degrees entered are converted with θ_rad = θ_deg × π ÷ 180 before any formula runs. This is distinct from a whole-circle calculator: it answers questions about a portion of the circle.

Worked Example: r = 10, θ = 60°

Convert: 60° = 60 × π ÷ 180 = π÷3 ≈ 1.047198 rad. Arc length s = rθ = 10 × 1.047198 ≈ 10.4720. Sector area = ½ × 10² × 1.047198 ≈ 52.3599 - exactly one-sixth of the full circle's π×100 ≈ 314.16, as expected since 60° is 1÷6 of 360°. Chord = 2 × 10 × sin(30°) = 20 × 0.5 = 10 (a 60° chord equals the radius - the equilateral-triangle fact). Triangle area = ½ × 100 × sin(60°) ≈ 43.301, so segment area = 52.360 − 43.301 ≈ 9.059 square units.

Where Arc & Sector Math Is Used

Engineering: gear teeth, cam profiles, pulley wrap angle, and pipe-bend developed length.

Architecture: arches, fan vaulting, curved staircases, and circular-segment window glazing area.

Navigation & astronomy: great-circle arc distance and angular size of celestial objects.

Data visualization: pie and donut charts - each slice is a sector with angle proportional to its share.

Common Mistakes & Edge Cases

Using degrees directly in s = rθ or A = ½r²θ - these formulas require radians; this tool converts for you, but the formulas themselves do not accept degrees.

Confusing the sector (radii + arc) with the segment (chord + arc); the segment is always the smaller of the two for θ < 180°.

Forgetting the half: sector area is ½r²θ, not r²θ.

Chord = 2r·sin(θ÷2), not r·sinθ - the half-angle is essential.

Angles ≥ 360° (2π) wrap the circle; results then exceed the full circle's area, which is geometrically meaningful only for spirals.

Related Concepts & Limits

The radian is the natural angle unit precisely because s = rθ and A = ½r²θ lose their constant factors only in radians; calculus derivatives of sine and cosine also assume radians. At θ = 2π the formulas reduce to the full circle: s = 2πr (circumference) and A = πr². For very small θ, sinθ ≈ θ, so the segment area ≈ ½r²(θ − sinθ) shrinks like θ³÷12 - the basis of small-angle approximations in optics and pendulum physics. This tool reports the slice as a percentage of the whole circle for quick sanity checks. Computations use double precision (≈ 15 significant digits); the calculator stays unit-agnostic, so label the answer in your chosen length unit.

Arc Length & Sector Area Calculator

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