How this instrument works
A straight angle measures π radians and, by much older convention, 180 degrees. Set those equal and this factor falls out unbidden: 1 rad = 180⁄π deg = 57.295779513082320877…, running on without pattern or end. Most conversion factors are decisions someone wrote into a standard, terminating wherever that person chose to stop. Geometry hands over this one instead, and it stops nowhere. Twelve figures, 57.2957795131, are that endless decimal cut where cutting does no harm.
Radians are what mathematics produces; degrees are what people read. Inverse trigonometric functions return radians, as do whole families of engineering results whose derivations never mention any angle unit at all. Almost nobody dimensions drawings, graduates dials or reports headings that way, so this direction — radian in, degree out — usually runs last, right before some figure reaches a screen, a print or a turret marking, rather than first as part of working something out.
Trades leaning on it are those where computed angles meet people or machine tools. Rover yaw leaves its attitude filter in radians and reaches its operator in degrees. Great-circle bearing formulas hand back radians while helmsmen want 0 to 360. Rotary axes in G-code — A, B and C — take their commands in degrees, though toolpath arithmetic behind them ran in radians throughout. Servo horns, theodolites, altazimuth mounts and rate tables all carry degree graduations, and spreadsheets keep DEGREES() around for precisely this handoff.
- Type your angle into Radians (rad). It opens on 3.141592653589793, what Python, JavaScript and C all print for π.
- Read your answer from Degrees (deg) directly beneath; it recomputes on every keystroke, with nothing submitted anywhere.
- Values below zero are refused, so add 6.283185307 to any negative atan2 result before typing it in.
- Spot-check against landmarks: 1.5707963268 rad returns 90 deg, and 1 rad returns 57.2957795131 deg.
- Working in milliradians? Divide by 1000 first, or convert as radians and shift your decimal point three places left.
Worked example — π radians from an attitude filter
Rover attitude filters report yaw as atan2 output, printing 3.141592653589793 rad — already loaded into Radians (rad) as this sheet's default. Degrees (deg) answers 180.0. That machine faces exactly opposite whichever axis it calls zero, which its driver, reading off north, calls due south.
Two small sins hide inside that clean 180.0. Decimal 3.141592653589793 is not π but its nearest double-precision neighbour, short by 1.2 × 10⁻¹⁶ — same residue that makes sin(π) return 1.2246467991473532e-16 instead of zero in every language you have tried. Our stored factor leans opposite, high by 1.8 × 10⁻¹¹. Multiplied together they yield 180.00000000006, over by two ten-millionths of one arcsecond and invisible at any sane display precision.
Scale past one full turn and nothing wraps: 7.5 rad returns 429.718346348 deg, or one revolution plus 69.718346348 deg. Subtract 360 yourself when readings must sit on compass cards.
Questions
How much error does 57.2957795131 carry?
About three parts in ten trillion, all of it on the high side. Since π never resolves into any finite decimal, neither does 180⁄π; 57.2957795131 is that quotient correctly rounded at twelve significant figures, which leaves it above truth by 1.8 × 10⁻¹¹. Swing an instrument through one entire revolution and everything accumulated stays under a millionth of an arcsecond, far beneath what encoders, theodolites or star trackers can resolve. Want it gone altogether? Multiply by 180/Math.PI or 180/math.pi and let your language supply π at machine precision.
Why is my angle negative, and why will the field not accept it?
Because atan2 answers within a half-open range of −π to π, anything pointing into lower half plane comes back signed, while this sheet takes zero or more. Add 6.283185307 — one full turn in radians — before entering such a figure. Geometry stays identical and your degree answer lands between 180 and 360, which is what compass cards, bearing reports and rotary tables expect. Values past one turn pass through untouched: 7 rad reads 401.070457 deg, not 41.070457.
How do I convert an angular rate in rad/s into deg/s or rpm?
Rates carry this same multiplier as angles do, since only their angular part changes units: 1 rad/s is 57.2957795131 deg/s, and angular acceleration follows suit, rad/s² becoming deg/s². Revolutions per minute need their own constant, folding in 2π radians per turn alongside 60 seconds per minute: 1 rad/s = 60⁄2π = 9.5492965855 rpm. So a spindle at 100π rad/s — near enough 314.159265 — is turning 18000 deg/s, which any machinist calls 3000 rpm.
What is one radian in degrees, minutes and seconds?
57° 17′ 44.81″. Start from decimal 57.295779513…, keep whole 57 degrees, multiply what remains — 0.295779513 — by 60 for 17.7467708 arcminutes, keep 17, then multiply that leftover 0.7467708 by 60 again for 44.806 arcseconds. Older surveying and astronomical tables print radians in exactly this sexagesimal form, sometimes truncated to 57° 17′ 45″.
Is 57.3 close enough for mental arithmetic?
For field estimates or sanity checks, comfortably. 57.3 sits high by 0.0074 percent, so right angles converted in your head land about 0.0066 deg — near enough 24 arcseconds — above true 90. Nobody folding sheet metal will ever see it. Surveyors closing traverses, machinists indexing rotary tables, or anyone chaining several conversions in sequence certainly will, and want all twelve figures this sheet carries.
Which quantities arrive in radians without ever saying so?
Every inverse trigonometric function — asin, acos, atan, atan2 — across C, Python, JavaScript, Java, Rust, Go, MATLAB and NumPy. Several standard engineering results are radians by construction too, even where no formula names any unit: twist of a loaded shaft, θ = TL⁄GJ; slope at the tip of a cantilever; phase angle behind power factor; divergence quoted for laser beams. Dimensions give it away. Anything built as one length over another, or as arc over its own radius, already counts in radians and needs only this multiplication to become readable.