Angle Converters — Degrees, Radians
Last updated:
Angle conversions span two dominant units that together cover every modern angular-measurement context from everyday navigation-and-engineering practice through advanced mathematics-and-physics work. The degree (°) is the universal practical angular unit, defined as 1/360 of a full circle and traceable to ancient Babylonian astronomy via the sexagesimal numeral system. The degree dominates navigation (latitude-and-longitude in degrees-arcminutes-arcseconds or decimal-degrees), surveying, mechanical-engineering, automotive-engineering, construction, geometry-and-trigonometry textbooks, and everyday angular measurements globally. The radian (rad) is the SI-derived mathematically natural angular unit, defined as the angle subtended at the centre of a circle by an arc length equal to the radius (1 full circle = 2π rad = 360°). The radian dominates mathematics, physics, and engineering work involving calculus-and-trigonometric-function calculations, where its dimensional-naturalness gives the simplest form for derivatives, integrals, Taylor series, and small-angle approximations. Cross-context angle conversion runs constantly at the boundaries between everyday-and-engineering practice (degrees) and advanced-mathematical-and-computational work (radians), with modern programming languages and computational frameworks defaulting to radians for trigonometric functions while engineering and surveying documentation stays in degrees. The conversion appears in computer-graphics work, GIS coordinate transformations, scientific-computing input/output formatting, robotics-and-control-systems, and physics-engineering simulation software cross-validation. The conversion is one of the most-run unit conversions in modern software-engineering, with every CAD-and-CAM, GPS-and-GIS, robotics, computer-graphics, and physics-simulation pipeline including degree-to-radian and radian-to-degree conversion at the human-input and human-display boundaries.
Units in this category
Degrees (°)
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.
Radians (rad)
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).
History of angle measurement
Angular measurement traces from ancient Babylonian astronomy through the modern era, with the degree-based system surviving for over four millennia and the radian emerging in the 1700s-1800s as the mathematically natural complement. The degree-arcminute-arcsecond (DMS) system traces to Babylonian sexagesimal-numeral-system astronomy (c. 2000-300 BCE), with 360 degrees per circle likely chosen because it is divisible by many small integers and approximates the days in a solar year. The Babylonian astronomers passed the convention to the Greeks (Hipparchus 2nd century BCE, Ptolemy 2nd century CE), then to medieval Arab astronomers, and through the Latin Almagest translation to medieval European astronomy. The radian was proposed informally by mathematicians in the 1700s and named "radian" by James Thomson in 1873, building on the recognition that arc-length-divided-by-radius gives the mathematically natural angular measure for calculus-and-trigonometric work. The 1960 SI specified the radian as a "supplementary unit" and the 1995 20th CGPM reclassified it as an SI-derived dimensionless unit. The degree retains universal practical dominance and is "accepted for use with the SI" by the BIPM SI brochure, while the radian dominates advanced mathematics-and-physics work where dimensional-naturalness matters.
Where angle conversions matter
Angle conversions appear across every modern navigation, surveying, engineering, computer-graphics, robotics, physics, and astronomy context. Navigation-and-geographic-positioning: every modern map, GPS coordinate, compass bearing, and aeronautical-navigation specification uses degrees, with latitude (-90° to +90°) and longitude (-180° to +180°) forming the universal geographic-coordinate system. Surveying-and-civil-engineering: bearing, deflection-angle, slope-angle, and grade-and-cant specifications in degrees on field-survey-and-construction documentation. Mechanical-engineering: bolt-circle angular-spacing, gear-tooth angles, bend-and-fold angles, machining-and-tooling angles in degrees. Automotive-engineering: camshaft timing-angle, ignition-timing-angle, suspension-camber-and-toe angles in degrees. Construction-and-architecture: roof-pitch-angles, stair-rise-angles, slope-and-grade specifications in degrees. Computer-graphics and computer-vision: rotation-and-transformation matrices, camera-projection mathematics use radians internally with degree input from designers-and-artists. Robotics-and-control-systems: joint-angles, rotation-matrices, quaternions all use radians internally with degree input from human operators. Physics-and-engineering computational work: angular-velocity (rad/s), angular-acceleration (rad/s²), angular-momentum, angular-frequency, wave-mechanics-and-quantum-mechanics phase-angles all use radians. Optics-and-laser-engineering: beam-divergence in milliradians-and-microradians. Firearms-optics: scope-windage-and-elevation in milliradians (mil) for tactical-shooting. Astronomy: arcseconds-and-microarcseconds for small-angle measurements with internal calculations in radians. The conversion runs at every cross-context engineering documentation boundary across everyday-engineering work (degrees) and advanced-mathematical-and-computational work (radians), with degree-to-radian and radian-to-degree conversion appearing in every CAD-and-CAM input pipeline, every GPS-and-GIS coordinate-transformation step, every robotics-control-system kinematics computation, and every physics-engineering simulation input-and-output boundary in modern software-engineering practice. The pairing supports the entire modern angular-measurement ecosystem from primary-school geometry through cutting-edge computational-physics-and-robotics work.
How to convert angle units
Angle-unit conversion runs against the relationship 360° = 2π rad exactly (a full circle), giving the conversion factors 1° = π/180 rad ≈ 0.0174533 rad and 1 rad = 180/π ° ≈ 57.29578°. The factors are exact in mathematics (containing π, an irrational constant) but are exact at any precision since π is mathematically defined rather than measured. For mental math: small-angle "° × 0.0175" lands on rad, "rad × 57.3" lands on °. Common reference values: 30° = π/6 rad ≈ 0.524 rad, 45° = π/4 rad ≈ 0.785 rad, 60° = π/3 rad ≈ 1.047 rad, 90° = π/2 rad ≈ 1.571 rad, 180° = π rad ≈ 3.1416 rad, 360° = 2π rad ≈ 6.2832 rad. Multiples and submultiples: degrees subdivide sexagesimally into arcminutes (' = 1/60°) and arcseconds (" = 1/3600°), used in DMS notation for navigation-and-astronomy; radians use SI prefix conventions with milliradian (mrad, 10⁻³ rad ≈ 0.0573°) and microradian (μrad, 10⁻⁶ rad) for small-angle work in firearms-optics, surveying, and laser-optics. The conversion is exact and universal across modern angular-measurement work, with the natural unit-scale appropriate for each application context.
All angle conversions
Frequently asked questions
How many radians are in 1 degree?
One degree equals π/180 radians, approximately 0.01745329 rad. The factor is exact in mathematics (containing π, an irrational constant) but is exact at any precision since π is mathematically defined. The relationship "360° = 2π rad" is the universal reference, giving the conversion factor π/180 for degree-to-radian and 180/π for radian-to-degree.
How many degrees are in 1 radian?
One radian equals 180/π degrees, approximately 57.29578°. The factor is the reciprocal of the degree-to-radian factor and is exact at any precision since π is mathematically defined. The "1 rad ≈ 57.3°" approximation appears in mental-math angular-velocity-and-rotation calculations where degrees-and-radians cross-references are needed.
Why do calculators have degree and radian modes?
Calculators support both modes because most real-world angular input is in degrees while mathematical and physics work uses radians. Trigonometric functions (sin, cos, tan) give different numerical results depending on the angular-mode setting — sin(90°) = 1 but sin(90 rad) ≈ 0.894. Always verify the angular-mode setting matches the input units before computing. Programming languages default to radians for trigonometric functions, requiring explicit conversion (e.g. Math.sin(angle * Math.PI / 180) for degree input).
What is a milliradian?
A milliradian (mrad) is one-thousandth of a radian, equal to about 0.0573° or 3.44 arcminutes. The unit is widely used in firearms-optics (scope-windage-and-elevation adjustments, the "mil" abbreviation), surveying (small-angle deflections), laser-optics (beam-divergence specifications), and astronomy (small-angle measurements). One milliradian gives a 1 metre offset per 1000 metres of distance — the convenient mental-math relationship that makes mrads useful for tactical-shooting and field-surveying work.
What is the relationship between rpm and rad/s?
Revolutions-per-minute (rpm) and radians-per-second (rad/s) are both angular-velocity units. The conversion is 1 rpm = 2π/60 rad/s ≈ 0.1047 rad/s. A typical industrial electric motor at 1800 rpm operates at 188.5 rad/s. Mechanical-engineering rotational-dynamics calculations (angular-acceleration, angular-momentum, bearing-life) typically convert rpm input to rad/s for the calculations and back to rpm for human-readable output.
Why are radians the natural unit for calculus?
Trigonometric function derivatives take their simplest form in radians: d/dx sin(x) = cos(x) holds only when x is in radians. In degrees, an extra factor of π/180 appears in every derivative, making degree-based calculus messy. The small-angle approximation sin(x) ≈ x also holds only for small x in radians (e.g. sin(0.01 rad) ≈ 0.01, but sin(0.01°) ≈ 0.000175). All advanced mathematics and physics work uses radians by mathematical necessity, with degree input converted at the input-and-output boundaries.
How precise should angle conversions be for engineering work?
For engineering work the angle conversion factor is exact (1° = π/180 rad), with the precision allowance coming from the underlying source-measurement precision. Most engineering documentation rounds to 6-7 significant figures (1° ≈ 0.0174533 rad, 1 rad ≈ 57.29578°), which is sufficient for typical mechanical, civil, and aerospace applications. Higher-precision applications (astronomy, geodesy, laser-pointing) preserve more digits, with arcseconds and microradians as the natural precision-units for sub-degree work.