How this instrument works
The degree is a leftover from Babylonian sexagesimal astronomy: split one turn into 360 parts and almost every useful fraction of that circle lands on whole numbers. The radian owes nothing to counting habits. Mark off an arc equal in length to the radius, join both ends of it to the centre, and that angle is one radian. Since circumference measures 2πr, one turn holds 2π radians, 360 deg equals 2π rad, and dividing through leaves 1 deg = π⁄180 rad ≈ 0.0174532925199.
Roger Cotes had this measure in hand by 1714, though his work reached print only after his death. Its name surfaced later still, in 1873, on an examination paper set by James Thomson at Queen's College Belfast — brother to Lord Kelvin. For decades afterwards, metrologists filed rad under “supplementary units”, a category neither base nor derived; Resolution 8 of the 20th CGPM scrapped that class in 1995 and listed rad among derived units, dimensionless, metre per metre. Which is why “rad” can quietly evaporate from an equation while “deg” never may.
Degrees still rule anywhere someone reads a dial: latitude and longitude, compass bearings, machinist protractors, CAD drawings, roof pitch, a solar panel's tilt. Radians rule anywhere calculus intrudes, since differentiating sin x yields cos x only if x counts radians. So Math.sin in JavaScript, C, Python, Java and Rust all expect radians, and spreadsheets ship a RADIANS() function precisely because so many people forget. Rifle optics and laser divergence specifications go one step smaller, quoting milliradians: 1 mrad subtends 10 cm at 100 m.
- Type your angle into the Degrees (deg) field — 180 is preloaded, since half turns are what most people check first.
- Read the Radians (rad) output beneath it; it recalculates on every keystroke, with no submit button involved.
- Sanity-check against landmarks you know: 90 deg should land on 1.570796 rad, and 57.29577951 deg on 1.000000 rad.
- Angles past one full turn convert happily — 720 deg returns 12.566371 rad — because nothing here wraps values into principal ranges.
- Going backwards? Divide your radian figure by 0.0174532925199, or multiply it by 57.29577951308.
Worked example — the half turn
Leave the Degrees (deg) field on its default of 180: a straight angle, due north swung round to due south. The Radians (rad) line answers 3.14159265359 — π to twelve significant figures. Half a turn is π radians by construction, so this particular row doubles as a check on that stored multiplier, which tracks π far past anything protractors, theodolites or shaft encoders could resolve.
Scale it and other landmarks stay legible. 60 deg gives 1.047198 rad, which is π⁄3; 45 deg gives 0.785398, or π⁄4; 1 deg gives 0.017453; and 57.29577951 deg comes back as 1.000000, the angle whose arc equals its own radius. Negative entries are refused rather than silently signed, so subtract from 360 first if you are working a bearing backwards.
Questions
Is 0.0174532925199 an exact conversion factor?
No, and no decimal ever could be. The true multiplier is π⁄180, and π is irrational, so any finite string of digits falls slightly short. 0.0174532925199 is π⁄180 rounded at twelve significant figures, low by roughly 2.5 parts in a trillion. Spread that over an Earth radius of 6371 km and a point shifts by about 16 micrometres — far below what any angular instrument resolves. Where you need better, most languages hand you π directly: multiply by Math.PI / 180 instead of a typed constant.
Why do trigonometric functions insist on radians?
Because calculus only comes out clean in that measure. Derivatives of sin x equal cos x, and sin x ≈ x for small x, exactly when x counts radians; in degrees both statements pick up a stray π⁄180. Standard libraries inherit that convention, which is why Math.sin(30) in JavaScript or Python returns about −0.988 — sine of 30 radians — rather than 0.5, which is what 30 degrees gives. Convert first, then call your function.
What does one radian actually look like?
Lay any circle's radius along its own rim as an arc, then join each end of that arc back to its centre. The wedge you have drawn opens by one radian, near enough 57.2957795 degrees — a slice of pie whose curved edge matches either straight edge in length. It follows that one full turn holds 2π ≈ 6.283 radians, a half turn holds π, and a square corner holds π⁄2 ≈ 1.5708.
Are the mils on a rifle scope the same as milliradians?
Not always, and that gap has cost people shots. A milliradian is one thousandth of a radian, so each turn holds 6283.19 of them, and 1 mrad covers 10 cm at 100 m — the basis of reticles marked MRAD. NATO artillery mils instead divide one turn into 6400, making each about 0.982 mrad; older Warsaw Pact optics used 6000. Establish which convention a device follows before trusting its graduations, especially when mixing turret clicks with reticle holdovers.
Why does the radian sometimes appear with no unit symbol at all?
Because it is dimensionless. A radian is arc length divided by radius, metre per metre, so units cancel and writing rad is optional — a convenience for readers rather than a requirement of algebra. The 20th CGPM confirmed this in 1995 by abolishing that old supplementary class. Degrees enjoy no such licence: drop the symbol from a degree figure and your reader will assume radians, which is among the commonest sources of silently wrong answers.
How do I convert radians back to degrees?
Multiply by 180⁄π ≈ 57.29577951308, or divide by 0.0174532925199; both routes agree to every digit shown here. Spreadsheets provide DEGREES() and RADIANS() so neither constant has to be typed by hand, and most programming languages expose π as a named constant for that same reason. This sheet runs one direction only — degrees in, radians out — so take the reciprocal yourself for a return trip.