Angle Converter
Degrees, radians, gradians, arcminutes, mrad — and the NATO mil.
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The engineering
Degrees rule drawings and navigation, radians rule every formula with a derivative in it, arcminutes rule optics and geodesy, and milliradians rule laser and beam work — 1 mrad of divergence is 1 m of spot growth per kilometer, the small-angle approximation earning its keep. Surveyors in parts of Europe still meet the gradian (100 per right angle) on older instruments and French plans.
The traps: the NATO mil is *not* a milliradian — armies rounded 2π×1000 = 6283.2 up to a clean 6400 per circle, so the two differ by 1.9% (and the Warsaw Pact used 6000, 4.7% off; a captured rangefinder lies fluently). The other classic is the degrees-mode calculator fed into sin(x) ≈ x reasoning, which only works in radians — spreadsheet trig defaults to radians, hand calculators often don't.
Where this math comes from
The 360° circle is Babylonian base-60 astronomy; Hipparchus of Nicaea, building trigonometry's first chord tables around 140 BCE, cemented degrees (and their sexagesimal minutes and seconds) into two millennia of practice. Roger Cotes worked out the natural, arc-length-based angle measure in 1714, but it went nameless until 1873, when James Thomson — the same Belfast professor who coined 'torque' — printed 'radian' on an exam paper.
The other two are institutional inventions: revolutionary France decimalized the right angle into 100 gradians in the 1790s alongside the metre (surveying instruments carried it across Europe), and artillerists adopted the milliradian for fire correction because at small angles, mils multiply by range to give meters. The armies then rounded it: 6400 per circle in NATO, 6000 in the Warsaw Pact — approximations chosen for gunnery arithmetic, not mathematics.
- 140 BCEHipparchus of NicaeaChord tables lock in the 360° Babylonian circle.
- 1714Roger CotesIdentifies the natural (radian) measure of angle.
- 1793French metric commissionGradian — the right angle decimalized to 100.
- 1873James ThomsonCoins 'radian' at Queen's College Belfast.
See the full timeline of the math behind every calculator →
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