Angle Unit Converter
Angles show up in many guises: degrees for everyday use, radians for calculus and physics, gradians (gon) for European surveying, turns for animation and gear ratios, arcminutes and arcseconds for astronomy, and NATO mils for military optics. Enter any one value and this tool instantly shows all 8 common angle units side-by-side, along with live sin / cos / tan and the principal value reduced into [0°, 360°) — handy for homework and engineering reference.
Please enter a valid number
Live value in every angle unit
Tip: 1 turn = 360° = 400 gon = 2π rad ≈ 6.2832 rad = 6400 mil (NATO). 1° = 60′ = 3600″. School trigonometry uses degrees; programming and physics default to radians.
Trigonometric values
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Principal value (in [0°, 360°)): —
- ° Everyday use, school maths, map bearings
- rad Calculus, physics, Math.sin / cos / tan in code
- gon European surveying (one right angle = 100 gon)
- turn Gear ratios, animation, phase (1 turn = 360°)
- ″ Astronomy, parallax, GPS precision
- mil Military optics, ballistic adjustment (6400 mil = 1 turn)
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Formula
Every unit is first converted into radians and then divided by the target unit's radian-equivalent. 1° = π/180 rad; 1 gon = π/200 rad; 1 turn = 2π rad.
- · 1 turn = 360° = 400 gon = 2π rad ≈ 6.2831853 rad = 6400 mil (NATO).
- · 1° = 60 arcminutes (′) = 3600 arcseconds (″) — sexagesimal subdivision.
- · 1 mrad = 0.001 rad ≈ 0.0573°. Widely used in optics, ballistics and scopes (1 mrad ≈ 10 cm at 100 m).
- · NATO mil = π/3200 rad, with 6400 mil per turn. Close in value to mrad but a different unit.
- · One right angle = 90° = 100 gon = π/2 rad — gradians give a decimal-friendly scale for surveying.
- · tan θ is mathematically undefined at 90°, 270° … (cos θ = 0); the calculator labels these "undefined".
- · Sources: BIPM SI Brochure 9th ed.; ISO 80000-3 (Space and time); NATO STANAG 2103 (mil definition).
Frequently asked
Does JavaScript's Math.sin() take degrees or radians?
Radians. JavaScript and almost every standard library (Python, C, Java, Swift …) take radians for their trig functions. To compute sin(30°) you write Math.sin(30 * Math.PI / 180) or Math.sin(Math.PI / 6). This tool does the conversion for you: enter 30°, see the equivalent radian value and the sin / cos / tan in one place — no mental arithmetic or copy-pasting code needed.
Radians vs degrees — which is more "natural"?
Radians. A radian is literally "arc length divided by radius" — a pure dimensionless ratio. Many fundamental results only work in radians: the small-angle approximation sin θ ≈ θ, the derivative d/dθ sin θ = cos θ, and the Taylor series for e^(iθ). Degrees are a Babylonian inheritance (360 ≈ days in a year, with many divisors) that suits everyday geometry but is just a convention. In calculus, complex numbers and physics, radians are the natural unit.
What's a gradian (gon)? Is it the same as "grad"?
Yes — gon, grad, gradian and grade all refer to the same unit. It's the decimal angle unit (100 gon per right angle, 400 gon per turn) promoted alongside the metric system during the French Revolution. Today it survives mainly in European surveying and cadastral work (Germany, France, Switzerland, Austria). It is handy for gradients — a 5 % slope equals 5 gon exactly. Scientific calculators with DEG / RAD / GRAD modes use "GRAD" for gon.
There are multiple "mil" units (mrad, NATO, Warsaw) — how do they differ?
Yes. "Mil" originally meant "divide a turn into 6400 parts," which is roughly 1 mrad (one thousandth of a radian) — but 2π × 1000 ≈ 6283 is not a round number, so several near-equivalents emerged: NATO mil = 1/6400 turn (artillery and optics, 1 mil ≈ 0.05625°); Warsaw Pact / Soviet mil = 1/6000 turn; Swedish mil = 1/6300 turn. This tool uses the NATO definition (STANAG 2103) and also provides the exact mrad. The optics industry standardises on mrad; militaries on NATO mil.
Why does the calculator say tan is "undefined" at 90°?
Because tan θ = sin θ ÷ cos θ, and cos θ = 0 at 90°, 270° (and generally any odd multiple of 90°). Division by zero is undefined. Some software prints a huge floating-point number like 1.6e16 there — that's a rounding artefact, not a real value of tan(90°). The calculator deliberately shows "undefined" instead of inventing a number.
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