Arc Length Calculator
Calculate arc length, sector area, and chord length from radius and central angle. Includes segment area, sagitta height, and sector perimeter.
Arc Length
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Sector Area —
Chord Length —
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Arc Length (s = rθ)
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Angle (radians) —
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Arc & Chord
Angle (radians) —
Arc Length —
Chord Length —
Segment
Segment Area —
Segment Height (sagitta) —
How to Use This Calculator
- Enter the Radius and Central Angle in degrees.
- Get arc length, sector area, and chord length instantly.
- Use the From Arc & Radius tab to find angle from arc length.
- Use the Sector tab to get sector perimeter.
- The Professional tab adds segment area and sagitta height.
Formula
Arc Length s = rθ | Sector Area = ½r²θ | Chord = 2r sin(θ/2)
Segment Area = Sector − Triangle = ½r²(θ − sinθ)
Example
r=10, θ=60°: Arc = 10×(π/3) ≈ 10.47 units, Sector = ½×100×(π/3) ≈ 52.36 units².
Frequently Asked Questions
- The arc length s of a circular arc is s = rθ, where r is the radius and θ is the central angle in radians. To convert degrees to radians: θ = degrees × π/180. For example, r = 10 and θ = 60°: θ_rad = 60 × π/180 = π/3 ≈ 1.0472 radians, arc length s = 10 × 1.0472 ≈ 10.47 units. For a full circle (θ = 360° = 2π radians), arc length = 2πr (the circumference). This calculator accepts degrees and converts automatically.
- The area of a circular sector (pie-slice region) is A = ½r²θ, where θ is in radians. For a 90° sector with r = 10: θ = π/2 radians, A = ½ × 100 × π/2 = 25π ≈ 78.54 square units. Equivalently, A = (θ/2π) × πr² = the fraction of the full circle times the full area. For 90° that is ¼ of the full area: πr²/4 = 25π ≈ 78.54. Use this formula for computing pizza slice areas, sprinkler coverage zones, and pie chart segments.
- A chord is the straight-line segment connecting two points on a circle. The chord length depends on the radius and the central angle: chord = 2r × sin(θ/2), where θ is the central angle in radians. For r = 10 and θ = 60°: chord = 2 × 10 × sin(30°) = 20 × 0.5 = 10 units. The longest possible chord is the diameter (when θ = 180°), with length = 2r. A chord divides the circle into two arcs: the minor arc (shorter) and the major arc (longer).
- A sector is the "pie slice" region bounded by two radii and the connecting arc — it includes the center of the circle. Its area = ½r²θ. A segment is the region between a chord and the arc, without including the center — like a crescent shape. Segment area = Sector area − Triangle area = ½r²θ − ½r²sin(θ) = ½r²(θ − sin θ). For example, at θ = π (180°), the segment is a semicircle (chord is the diameter) and area = ½r²(π − 0) = πr²/2.
- The sagitta (or "arrow") is the maximum height of a circular segment — the perpendicular distance from the midpoint of the chord to the arc at its highest point. Formula: sagitta = r(1 − cos(θ/2)), where r is the radius and θ is the central angle in radians. For r = 10 and θ = 60°: sagitta = 10(1 − cos 30°) = 10(1 − 0.866) ≈ 1.34 units. A sagitta close to r means a nearly complete circle; a sagitta near 0 means a very shallow arc. Sagittas are used in optics (lens curvature) and arch construction.