Radians to Degrees (rad to °)

The radian (rad) is defined as the angle subtended at the centre of a circle by an arc length equal to the radius. Equivalently, one radian is the angle for which arc-length divided by radius equals 1. The full circle subtends 2π radians (the circumference 2πr divided by the radius r). The relationship to degrees is 360° = 2π rad exactly, hence 1 rad = 180/π ° ≈ 57.29578°. The radian is dimensionless (the ratio of two lengths), classified as an SI-derived dimensionless unit by the 1995 20th CGPM. The radian is the mathematically natural angular unit because trigonometric-function derivatives take their simplest form in radians: d/dx sin(x) = cos(x), d/dx cos(x) = -sin(x), and the small-angle approximation sin(x) ≈ x holds for small x in radians. The milliradian (mrad, 10⁻³ rad ≈ 0.0573°) is widely used in firearms-optics, surveying, and astronomy for small-angle work. The microradian (μrad) appears in laser-and-optics precision-pointing applications.

The radian as an angular unit was proposed by mathematicians and physicists in the 1700s and formalised in the late 1800s as the mathematically natural angular unit for calculus-and-physics work. The name "radian" was coined by James Thomson (brother of Lord Kelvin) in 1873, building on earlier informal usage of arc-length-divided-by-radius as the natural angular measure. Mathematicians had long recognised that derivatives and integrals of trigonometric functions take their simplest form when angles are measured in radians (d/dx sin(x) = cos(x) holds only when x is in radians; in degrees, an extra factor of π/180 appears). The radian was formally adopted by the International Bureau of Weights and Measures (BIPM) and recognised as an SI-coherent supplementary unit by the 1960 11th CGPM and reclassified as an SI-derived dimensionless unit by the 1995 20th CGPM. The radian dominates mathematics, physics, and engineering work involving trigonometric calculations, angular-velocity, angular-momentum, and angular-frequency specifications, while the degree retains its universal dominance in everyday and applied engineering practice. Modern programming languages (JavaScript Math.sin, Python math.sin, C++ std::sin, etc.) implement trigonometric functions in radians by default, with degree-to-radian conversion required for everyday-engineering input.

Mathematics, physics, and engineering work involving trigonometric-and-angular calculations. Calculus-and-trigonometric-function work in mathematics: derivatives, integrals, Taylor series, and Fourier analysis all use radians by mathematical necessity. Physics: angular-velocity (ω in rad/s), angular-acceleration (α in rad/s²), angular-momentum, angular-frequency (in rad/s for oscillators-and-rotating-systems), wave-mechanics-and-quantum-mechanics phase angles. Mechanical-engineering rotational-dynamics: angular-velocity specifications for shafts, motors, and turbines (often converted from rpm to rad/s for dynamics calculations: 1 rpm = 2π/60 rad/s ≈ 0.1047 rad/s). Electrical-engineering AC analysis: phase angles in radians for sinusoidal voltages, currents, and impedances. Robotics-and-control-systems: joint-angles, rotation-matrices, quaternions all use radians internally. Computer-graphics and computer-vision: rotation-and-transformation matrices, camera-projection mathematics use radians. Optics-and-laser-engineering: beam-divergence in milliradians-and-microradians. Firearms-optics: scope-windage-and-elevation-adjustments in milliradians (mil) for tactical-shooting work. Astronomy-and-astrophysics: small-angle measurements in arcseconds-and-microarcseconds, with internal calculations in radians. Modern programming languages and computational frameworks default to radians for trigonometric functions.

The degree (°) is defined as 1/360 of a full circle (a complete rotation). Equivalently, the degree is π/180 radians ≈ 0.01745329 radians, derived from the radian definition (1 full circle = 2π radians = 360°). The relationship is exact in mathematics, with the degree-to-radian factor irrational (containing π) but exact at any precision. The degree is subdivided sexagesimally into 60 arcminutes (') per degree and 60 arcseconds (") per arcminute (1° = 60' = 3600"), giving the degree-arcminute-arcsecond (DMS) notation used in navigation, astronomy, and surveying (e.g. 40° 26' 28" N latitude). Modern decimal-degree notation (40.4411° N) is increasingly used in computer-and-GIS contexts, with the relationship between DMS and decimal-degree being 40° 26' 28" = 40 + 26/60 + 28/3600 = 40.4411°. The degree is "accepted for use with the SI" by the BIPM SI brochure despite not being SI-coherent, given its universal practical importance in navigation, astronomy, surveying, engineering, and everyday geometry.

The degree as an angle unit traces to ancient Babylonian astronomy, where the sexagesimal (base-60) numeral system led astronomers to divide the full circle into 360 equal parts — likely chosen because 360 is divisible by many small integers (1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360) and approximates the 365.25 days in a solar year. The Babylonian astronomers passed the convention to the Greeks (notably Hipparchus and Ptolemy in the second century BCE and second century CE respectively), then to medieval Arab astronomers, and through the Latin Almagest translation to medieval European astronomy. The subdivision of the degree into 60 arcminutes (') and the arcminute into 60 arcseconds (") preserved the Babylonian sexagesimal subdivision, with the convention surviving through Renaissance and modern astronomy. The degree-arcminute-arcsecond system remained the universal angular measurement convention for navigation (latitude-and-longitude), astronomy, surveying, geometry, and engineering through the modern era. The radian was proposed in the 1700s and formalised in the late 1800s as the more mathematically natural angular unit, but the degree retains overwhelming dominance in everyday and applied engineering practice. The 1960 SI specifies the radian as the SI-coherent angular unit, but the degree remains "accepted for use with the SI" by the BIPM SI brochure given its universal practical importance.

Navigation and geographic-positioning globally — every modern map, GPS coordinate, compass bearing, and aeronautical-navigation specification uses degrees. Latitude (-90° to +90° N/S from equator) and longitude (-180° to +180° E/W from Greenwich) form the universal geographic-coordinate-system. Compass bearings (0° to 360° clockwise from north) for navigation, surveying, and aviation. Astronomy: celestial-coordinate-systems (right-ascension typically in hours-minutes-seconds, declination in degrees-arcminutes-arcseconds), ecliptic-coordinates, galactic-coordinates. Surveying-and-civil-engineering: bearing, deflection-angle, slope-angle, and grade-and-cant specifications in degrees. Mechanical-engineering: bolt-circle angular-spacing, gear-tooth angles, bend-and-fold angles, machining-and-tooling angles. Automotive-engineering: camshaft timing-angle, ignition-timing-angle, suspension-camber-and-toe angles. Construction-and-architecture: roof-pitch-angles, stair-rise-angles, slope-and-grade specifications. Geometry-and-trigonometry textbooks at primary, secondary, and undergraduate level, with the radian appearing primarily in advanced calculus-and-physics work where mathematical naturalness matters. The degree is the universal practical angular unit, with the radian dominating mathematical-and-physics contexts where dimensional consistency requires it.