{
  "id": "arc-length",
  "version": "21edb6e60a70",
  "status": "published",
  "name": "Arc Length Calculator",
  "question": "What is the arc length?",
  "summary": "Finds the arc length s = rθ of a circle from the radius and the central angle, or any two of the radius, angle, arc length, and sector area, with the chord.",
  "category": "math",
  "subcategory": "geometry",
  "url": "https://www.acalculator.org/math/arc-length-calculator",
  "markdown": "https://www.acalculator.org/math/arc-length-calculator.md",
  "kind": "formulas",
  "method": "s = rθ, where θ is the central angle in radians; the sector area is A = ½r²θ = ½sr.",
  "assumptions": [
    "The central angle is more than 0° and at most 360°. It can be typed in degrees or radians; the formulas use radians (θ in radians = degrees × π ÷ 180).",
    "The radius, arc length, and sector area are positive numbers in any one unit; the area is in that unit squared.",
    "An arc longer than the circumference, or a sector bigger than the circle, has no answer."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "r": {
        "title": "Radius",
        "description": "The radius r of the circle: from the centre to the arc.",
        "type": "number",
        "exclusiveMinimum": 0,
        "maximum": 1e+100
      },
      "angle": {
        "title": "Central angle",
        "description": "The angle θ at the centre of the circle between the two radii that end the arc, up to 360°.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "angle",
        "exclusiveMinimum": 0,
        "maximum": 6.283185307179586
      },
      "s": {
        "title": "Arc length",
        "description": "The length s of the arc: the part of the circle between the two radii.",
        "type": "number",
        "exclusiveMinimum": 0,
        "maximum": 1e+101
      },
      "area": {
        "title": "Sector area",
        "description": "The area A of the sector: the slice of the circle that the arc bounds.",
        "type": "number",
        "exclusiveMinimum": 0,
        "maximum": 1e+201
      }
    }
  },
  "solve": {
    "fillAny": 2,
    "variables": [
      "r",
      "angle",
      "s",
      "area"
    ],
    "inputsOnly": []
  },
  "outputs": {
    "r": {
      "label": "Radius",
      "description": "The radius r of the circle: from the centre to the arc.",
      "format": "number"
    },
    "theta": {
      "label": "Central angle",
      "description": "The angle θ at the centre of the circle between the two radii that end the arc.",
      "format": "quantity",
      "unit": "deg"
    },
    "s": {
      "label": "Arc length",
      "description": "The length s of the arc: the part of the circle between the two radii.",
      "format": "number"
    },
    "a": {
      "label": "Sector area",
      "description": "The area A of the sector: the slice of the circle that the arc bounds.",
      "format": "number"
    },
    "chord": {
      "label": "Chord length",
      "description": "The straight line between the two ends of the arc: 2r sin(θ/2).",
      "format": "number"
    },
    "perimeter": {
      "label": "Sector perimeter",
      "description": "The distance around the sector: the arc plus two radii, s + 2r.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "r": 10,
      "angle": 1.0471975511965976
    },
    "outputs": {
      "s": 10.471975511965976,
      "a": 52.35987755982988,
      "chord": 9.999999999999998,
      "perimeter": 30.471975511965976
    },
    "text": "An arc with radius 10 and a central angle of 60° is 10.471976 long, and its sector has an area of 52.359878."
  },
  "examples": [
    {
      "given": {
        "r": 10,
        "angle": 1.0471975511965976
      },
      "expect": {
        "s": 10.471975511965976,
        "area": 52.35987755982988,
        "chord": 9.999999999999998,
        "perimeter": 30.471975511965976
      },
      "source": "OpenStax, Precalculus 2e, §5.1 Angles (https://openstax.org/books/precalculus-2e/pages/5-1-angles); hand calculation in content.mdx: s = 10 × π/3, A = ½ × 100 × π/3, chord = 20 sin 30° = 10; Python 3: 10*math.pi/3, 0.5*100*math.pi/3, 20*math.sin(math.pi/6)"
    },
    {
      "given": {
        "r": 5,
        "s": 12
      },
      "expect": {
        "angle": 2.4,
        "area": 30,
        "chord": 9.320390859672262
      },
      "source": "hand calculation in content.mdx: θ = 12 ÷ 5 = 2.4 rad, A = ½ × 12 × 5 = 30; Python 3: 10*math.sin(1.2)"
    },
    {
      "given": {
        "s": 15,
        "area": 60
      },
      "expect": {
        "r": 8,
        "angle": 1.875,
        "chord": 12.897297732171088
      },
      "source": "hand calculation in content.mdx: r = 2A ÷ s = 8, θ = s ÷ r = 1.875 rad; Python 3: 16*math.sin(1.875/2)"
    },
    {
      "given": {
        "angle": 1.5707963267948966,
        "area": 314.1592653589793
      },
      "expect": {
        "r": 20,
        "s": 31.41592653589793,
        "chord": 28.2842712474619
      },
      "source": "hand calculation in content.mdx: r = √(2A ÷ θ) = √400 = 20, s = 20 × π/2 = 10π, chord = 40 sin 45° = 20√2; Python 3: 40*math.sin(math.pi/4)"
    },
    {
      "given": {
        "r": 3,
        "angle": 6.283185307179586
      },
      "expect": {
        "s": 18.84955592153876,
        "area": 28.274333882308138,
        "chord": 0
      },
      "source": "hand calculation in content.mdx: a full turn is the whole circle, s = 2π × 3 = 6π, A = π × 9 = 9π"
    }
  ],
  "sources": [
    "OpenStax, Precalculus 2e, §5.1 Angles (arc length s = rθ, area of a sector A = ½θr²). https://openstax.org/books/precalculus-2e/pages/5-1-angles",
    "NIST Digital Library of Mathematical Functions, §4.14 (sine) and §3.12 Mathematical Constants (π). https://dlmf.nist.gov/3.12"
  ],
  "related": [
    "circle",
    "circumference",
    "area-of-a-circle",
    "triangle"
  ],
  "changelog": []
}