{
  "id": "angle",
  "version": "f7b931a0f9f3",
  "status": "published",
  "name": "Angle Calculator",
  "question": "What is this angle in other units?",
  "summary": "Converts an angle between degrees, radians, multiples of π, gradians, turns and degrees-minutes-seconds, and finds its coterminal and reference angles, quadrant, type, complement, supplement, and sine, cosine and tangent.",
  "category": "math",
  "subcategory": "geometry",
  "url": "https://www.acalculator.org/math/angle-calculator",
  "markdown": "https://www.acalculator.org/math/angle-calculator.md",
  "kind": "function",
  "method": "radians = degrees × π ÷ 180; gradians = degrees × 10 ÷ 9; turns = degrees ÷ 360; coterminal = degrees mod 360; reference angle from the quadrant; complement = 90° − angle; supplement = 180° − angle.",
  "assumptions": [
    "An angle typed in degrees, multiples of π, gradians or turns is converted exactly; one typed in radians uses π to double precision.",
    "Negative angles turn clockwise; coterminal angles differ by whole turns of 360°.",
    "The complement is shown for angles from 0° to 90°, the supplement for angles from 0° to 180°, and the type for angles from 0° to 360°.",
    "Seconds are rounded half up to 2 decimal places, carrying into minutes and degrees."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "x": {
        "title": "Angle",
        "description": "The angle, in the unit picked below; negative angles turn clockwise.",
        "type": "number",
        "minimum": -1000000000,
        "maximum": 1000000000
      },
      "unit": {
        "title": "Unit",
        "description": "The unit of the angle: degrees, radians, multiples of π radians, gradians, or turns.",
        "type": "string",
        "enum": [
          "deg",
          "rad",
          "pi",
          "grad",
          "turn"
        ]
      }
    }
  },
  "outputs": {
    "degrees": {
      "label": "Degrees",
      "description": "The angle in degrees.",
      "format": "number"
    },
    "radians": {
      "label": "Radians",
      "description": "The angle in radians: degrees × π ÷ 180.",
      "format": "number"
    },
    "piRadians": {
      "label": "Radians as a multiple of π",
      "description": "The angle in radians divided by π: degrees ÷ 180.",
      "format": "number"
    },
    "gradians": {
      "label": "Gradians",
      "description": "The angle in gradians: degrees × 10 ÷ 9.",
      "format": "number"
    },
    "turns": {
      "label": "Turns",
      "description": "The angle in full turns: degrees ÷ 360.",
      "format": "number"
    },
    "dms": {
      "label": "Degrees, minutes, seconds",
      "description": "The angle as whole degrees, minutes and seconds, the seconds rounded half up to 2 decimals.",
      "format": "text"
    },
    "coterminal": {
      "label": "Coterminal angle in [0°, 360°)",
      "description": "The angle plus or minus whole turns, between 0° and 360°.",
      "format": "number"
    },
    "reference": {
      "label": "Reference angle",
      "description": "The acute angle between the terminal side and the x-axis, from 0° to 90°.",
      "format": "number"
    },
    "quadrant": {
      "label": "Quadrant",
      "description": "The quadrant or axis of the terminal side.",
      "format": "text"
    },
    "kind": {
      "label": "Type",
      "description": "Zero, acute, right, obtuse, straight, reflex, or full turn, for angles from 0° to 360°.",
      "format": "text"
    },
    "complement": {
      "label": "Complement",
      "description": "90° minus the angle, for angles from 0° to 90°.",
      "format": "number"
    },
    "supplement": {
      "label": "Supplement",
      "description": "180° minus the angle, for angles from 0° to 180°.",
      "format": "number"
    },
    "sin": {
      "label": "Sine",
      "description": "The sine of the angle.",
      "format": "number"
    },
    "cos": {
      "label": "Cosine",
      "description": "The cosine of the angle.",
      "format": "number"
    },
    "tan": {
      "label": "Tangent",
      "description": "The tangent of the angle: sine ÷ cosine; none where the cosine is 0 (90°, 270°).",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "x": 150,
      "unit": "deg"
    },
    "outputs": {
      "degrees": 150,
      "radians": 2.6179938779914944,
      "piRadians": 0.8333333333333334,
      "gradians": 166.66666666666666,
      "turns": 0.4166666666666667,
      "dms": "150° 0′ 0″",
      "coterminal": 150,
      "reference": 30,
      "quadrant": "Quadrant II",
      "kind": "Obtuse",
      "supplement": 30,
      "sin": 0.49999999999999994,
      "cos": -0.8660254037844387,
      "tan": -0.5773502691896256
    },
    "text": "150° is 2.617993878 radians."
  },
  "examples": [
    {
      "given": {
        "x": 150,
        "unit": "deg"
      },
      "expect": {
        "radians": 2.6179938779914944,
        "piRadians": 0.8333333333333334,
        "gradians": 166.66666666666666,
        "dms": "150° 0′ 0″",
        "coterminal": 150,
        "reference": 30,
        "quadrant": "Quadrant II",
        "kind": "Obtuse",
        "supplement": 30,
        "sin": 0.5,
        "cos": -0.8660254037844387,
        "tan": -0.5773502691896256
      },
      "source": "OpenStax Algebra and Trigonometry 2e, section 7.1, Angles (radians = degrees × π ÷ 180; coterminal and reference angles), CC BY 4.0 (https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-1-angles, retrieved 2026-10-01); hand calculation in content.mdx: 150 × π ÷ 180 = 5π ÷ 6",
      "tolerance": 1e-12
    },
    {
      "given": {
        "x": -405,
        "unit": "deg"
      },
      "expect": {
        "coterminal": 315,
        "reference": 45,
        "quadrant": "Quadrant IV",
        "turns": -1.125,
        "dms": "−405° 0′ 0″"
      },
      "source": "OpenStax Algebra and Trigonometry 2e, section 7.1, Angles: coterminal angles differ by multiples of 360° (https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-1-angles); hand calculation in content.mdx: −405 + 2 × 360 = 315"
    },
    {
      "given": {
        "x": 12.5125,
        "unit": "deg"
      },
      "expect": {
        "dms": "12° 30′ 45″",
        "complement": 77.4875,
        "kind": "Acute"
      },
      "source": "NIST Special Publication 811, Appendix B: 1° = 60′, 1′ = 60″ (https://www.nist.gov/pml/special-publication-811); hand calculation in content.mdx: 0.5125 × 60 = 30.75′, 0.75 × 60 = 45″"
    },
    {
      "given": {
        "x": 0.5,
        "unit": "pi"
      },
      "expect": {
        "degrees": 90,
        "quadrant": "On the positive y-axis",
        "kind": "Right",
        "sin": 1,
        "cos": 0,
        "complement": 0
      },
      "source": "OpenStax Algebra and Trigonometry 2e, section 7.1, Angles: π ÷ 2 radians = 90° (https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-1-angles); hand calculation in content.mdx"
    },
    {
      "given": {
        "x": 1,
        "unit": "rad"
      },
      "expect": {
        "degrees": 57.29577951308232,
        "radians": 1,
        "dms": "57° 17′ 44.81″"
      },
      "source": "NIST Special Publication 811, Appendix B: 1 rad = 57.295 78° (https://www.nist.gov/pml/special-publication-811); hand calculation in content.mdx: 180 ÷ π = 57.2957795°",
      "tolerance": 1e-12
    }
  ],
  "sources": [
    "OpenStax, Algebra and Trigonometry 2e, section 7.1, Angles (degrees and radians, coterminal angles, reference angles), CC BY 4.0, retrieved 2026-10-01. https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-1-angles",
    "NIST Special Publication 811, Guide for the Use of the International System of Units, Appendix B (degree, minute, second, gon, revolution). https://www.nist.gov/pml/special-publication-811"
  ],
  "related": [
    "trigonometry",
    "right-triangle",
    "triangle",
    "law-of-cosines"
  ],
  "changelog": []
}