Angle Unit Converter (Degrees · Radians · Gradians · DMS)
Convert any angle between degrees, radians, gradians, turns, milliradians, arcminutes, arcseconds and DMS, with a live SVG preview and π-fraction output.
Conversion details
Internal canonical unit: turns. Conversion factors per turn: 360 deg, 2π rad, 400 grad, 2000π mrad, 21600 arcmin, 1 296 000 arcsec. DMS input accepts spaces or °/′/″ symbols; output is formatted with two decimals on the seconds. The π-fraction recognizer snaps to p·π/q within 1e-6 (p ≤ 8q, q ≤ 24).
How to use
- Type a value in the input box — pick the unit from the dropdown.
- All other units update instantly with appropriate precision per unit.
- Use the quick chips for canonical angles (30°, 45°, 90°, π, π/2…) or to seed an experiment.
- DMS input accepts spaces or °/′/″ symbols — e.g. "37° 25' 19.07\"".
- Click the copy icon on any row to copy that representation.
About this tool
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Most code, calculators, and CAD tools speak different angle dialects. Trigonometry and JavaScript's Math.* default to radians; everyday geometry and CAD use degrees; surveyors and European engineers use gradians (400 to a full turn — chosen so a right angle is exactly 100 grad); shooters and ballistics use milliradians (one mrad ≈ 10 cm at 100 m); astronomers and cartographers carry sub-degree precision in arcminutes (1/60°) and arcseconds (1/3600°), often written as DMS (Degrees-Minutes-Seconds, e.g. 37° 25' 19.07"). This tool converts between all of them in one shot. Enter any value in any unit — the output panel shows the same angle in every other unit, recognizes common π-fractions (π/6, π/4, π/3, π/2, π, etc.) when they match within 1e-6, and renders an SVG circle preview measuring counter-clockwise from the top (standard math convention reflected as a 12-o'clock zero for legibility). DMS parsing is forgiving: '37° 25 19.07', '37 25 19', and '37.42196°' all parse correctly.
Frequently asked questions
What's a gradian, and why does it exist?
A gradian (also gon) divides a full turn into 400 equal parts instead of 360. It was introduced during the French Revolution alongside the metric system. The appeal: a right angle is exactly 100 grad, so percent-slope on a road or roof is the gradian of its incline — no calculator needed. Gradians never displaced degrees in everyday use but survive in European surveying, geodesy, and some artillery / civil-engineering software.
What's a milliradian (mrad / mil)?
1 mrad = 0.001 radian. At distance r, an arc of one mrad subtends r/1000, so one mrad ≈ 10 cm at 100 m, ≈ 1 m at 1 km — extremely handy for range estimation. NATO uses a 6400-mil 'angular mil' which is close but not equal to a true mathematical mrad (2π × 1000 ≈ 6283.19 true mrad). This converter uses the mathematical mrad; for NATO mils, multiply mrad by 6400 / (2π × 1000) ≈ 1.01859.
How is DMS parsed and formatted?
Input accepts any of: '37.42196', '37 25 19.07', '37° 25′ 19.07″', or '37° 25\'19.07"' — degree / minute / second symbols are stripped and the remaining numbers are parsed positionally. Negative values are recognized from a leading minus on the degrees field. Output is formatted as 'D° M\' S.SS"' with two decimal places on the seconds, which is enough for ~3 cm precision at Earth-surface latitudes.
Why does the π-fraction badge sometimes not appear?
We only display a π-fraction badge when the angle matches p·π/q within 1e-6, with p ≤ 8q and q ≤ 24. So π/4, 3π/2, 7π/12 will all snap, but 1.2345 rad will not because no nearby small fraction is within tolerance. This avoids cluttering the UI with bogus 'closest' fractions for non-special angles.
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