Multiply by π/180 ≈ 0.017453: degrees × π/180 = radians. Examples: 90° = π/2 ≈ 1.5708 rad; 180° = π ≈ 3.1416 rad; 360° = 2π ≈ 6.2832 rad; 45° = π/4 ≈ 0.7854 rad. Reverse (radians to degrees): multiply by 180/π ≈ 57.2958. Example: 1 rad = 57.296°. Radians are the "natural" angle unit because the arc length formula simplifies to s = rθ (no conversion factor) and all trigonometric derivatives are correct without extra constants.

Multiply by 10/9: degrees × 10/9 = grad. Example: 90° = 100 grad exactly; 180° = 200 grad; 360° = 400 grad. Reverse: multiply by 9/10. Example: 300 grad = 270°. The key attraction of gradians is that a right angle is exactly 100 grad — making decimal angular arithmetic natural. A 1-grad step = 0.9°. Surveyors in France, Germany, and Scandinavia commonly use grad (also called gon) in theodolite readings and cadastral maps.

1 degree = 60 arcminutes (′). degrees × 60 = arcminutes. Example: 1.5° = 90 arcminutes. Reverse: arcminutes ÷ 60 = degrees. Example: 30′ = 0.5°. The arcminute traces back to Babylonian astronomy. Its most practical modern connection: 1 arcminute of latitude ≈ 1 nautical mile (1,852 m) — which is literally how the nautical mile was defined. GPS coordinates in DMS (degrees-minutes-seconds) use arcminutes as the middle component of the position string.

Formula: DD = degrees + (minutes/60) + (seconds/3600). Example: 48° 51′ 29″ = 48 + (51/60) + (29/3600) = 48.8581° (decimal degrees). Reverse: degrees = integer part; minutes = fractional degrees × 60 (take integer); seconds = remaining fraction × 60. The DMS format is used in paper maps, GNSS receivers, and aviation charts; decimal degrees (DD) is preferred in GIS software, APIs, and spreadsheet calculations. The converter outputs both formats simultaneously.

Divide by 360: degrees ÷ 360 = revolutions. Example: 180° = 0.5 rev; 720° = 2 rev; 90° = 0.25 rev. Reverse: revolutions × 360 = degrees. Revolution, turn, and circle are identical units — all represent one complete 360° rotation. They're used in RPM (revolutions per minute), gear ratio calculations, and motor specifications. A motor spinning at 3,000 RPM completes 3,000 × 360 = 1,080,000 degrees of rotation per minute.

The NATO mil divides a full circle into 6,400 mils. Mil to degrees: mil × 0.05625 = degrees. Example: 1,600 mil = 90°. Degrees to mil: degrees × 160/9 ≈ 17.778. Example: 45° = 800 mil. The mil's defining property: at 1,000 m range, 1 mil = 1 m lateral span — making target width estimation formulaic: mils × range/1,000 = size in metres. Note: Soviet/Russian forces use a 6,000-mil circle (1 mil = 0.06°); NATO uses 6,400. These are incompatible systems.

Radians are natural because they make calculus clean. The derivative of sin(x) is cos(x) only when x is in radians — in degrees, a factor of π/180 appears in every derivative. Similarly, the Taylor series sin(x) ≈ x − x³/6 + x⁵/120… is exact only in radians. The arc length formula s = rθ requires no conversion factor with radians. For these reasons, all scientific computing languages (Python, MATLAB, C++, Julia) use radians internally by default — any angle you pass to sin(), cos(), or tan() must be in radians unless you explicitly use a degree variant.

Arcminutes to degrees: arcmin ÷ 60 = degrees. Example: 90′ = 1.5°. Arcseconds to degrees: arcsec ÷ 3,600 = degrees. Example: 3,600″ = 1°. To convert 0°45′30″: 0 + (45/60) + (30/3600) = 0.7583°. For astronomy: the full Moon's diameter is ~30 arcminutes = 0.5°; Jupiter's disc at opposition is ~50 arcseconds = 0.0139°; Hubble's resolution limit is ~0.05 arcseconds = 0.0000139°. These tiny angles require arcsecond precision — decimal degrees would require 6+ decimal places.

Three steps: (1) Enter any angle value — decimals work (e.g., 45.5°, 1.5708 rad, 100 grad, 6400 mil, 0.5 rev). (2) Select the source unit from the dropdown: degree, radian, grad/gon, arcminute, arcsecond, mil, revolution, turn, circle, quadrant, right angle, sign, or sextant. (3) All units update instantly. Tap Copy beside any row to place the value in your clipboard. Free at untangletools.com/unit/category/angle — no login, no ads during conversion.

Yes — decimal inputs are fully supported. Enter 33.333 for degrees; enter 1.0472 for π/3 radians (≈ 60°). The converter uses 64-bit floating-point precision throughout. For irrational inputs like π: enter the decimal approximation (3.14159265). Examples: 33.333° = 0.5818 rad = 37.037 grad; 0.785398 rad = 45°. The DMS component breaks down the decimal degree output into degrees, arcminutes, and arcseconds for geographic coordinate use.

Yes — all of them. Right angle = 90°; quadrant = 90° (they are equivalent — a circle has 4 quadrants of 90° each). Sign = 30° (12 signs = 360°, mapping to zodiac signs). Sextant = 60° (6 sextants = 360°). Circle = revolution = turn = 360° exactly. These units appear in historical texts, astrology, celestial navigation, and legacy engineering — the converter handles all of them with the same one-click-copy interface as degrees and radians.

Yes. All factors use exact mathematical definitions: 1 radian = 180/π degrees (exact, since π is defined); 1 grad = 9/10 degrees (exact); 1 mil = 360/6,400 = 0.05625° (exact); 1 revolution = 360° (exact); 1 arcminute = 1/60° (exact); 1 arcsecond = 1/3,600° (exact). Results use full 64-bit floating-point — no intermediate rounding. Completely free at untangletools.com/unit/category/angle with no account or subscription required.

GPS coordinates appear in three formats: Decimal Degrees (DD) — e.g., 48.8566°N, 2.3522°E (used by Google Maps, APIs); Degrees Decimal Minutes (DDM) — e.g., 48° 51.396′N (used by most GPS units); Degrees Minutes Seconds (DMS) — e.g., 48° 51′ 23.76″N (used on paper maps and aviation charts). Converting between them: DD = D + M/60 + S/3600. Precision: 1° latitude ≈ 111 km; 1′ ≈ 1.85 km; 1″ ≈ 30 m; 0.0001° ≈ 11 m. For precision surveying, arcsecond-level accuracy is essential.

Surveying favours gradians because: (1) A right angle = exactly 100 gon — trivially easy to check perpendicularity. (2) A full circle = 400 gon — decimal division is cleaner than 360°. (3) Angular arithmetic (adding/subtracting bearings) avoids the 60-base system of degrees/minutes/seconds. In a traverse survey, adjusting misclosure is simpler in decimal gon than in DMS. European theodolites (particularly Leica and Zeiss instruments) default to gon readouts. The ISO 80000-3 standard recognises both degree and radian but not grad — yet grad remains dominant in European geodesy and cadastral surveying.

A parsec (pc) is defined as the distance at which 1 Astronomical Unit (Earth-Sun distance) subtends an angle of exactly 1 arcsecond. This makes stellar parallax distances automatic: a star with 0.5 arcsec parallax is 2 parsecs away. Practical examples: the nearest star (Proxima Centauri) is 1.295 pc = 4.24 light-years; its parallax is 0.772 arcseconds. Hubble Space Telescope achieves ~0.05 arcsecond resolution; the Event Horizon Telescope resolved M87's black hole at ~20 microarcseconds (0.00002″). At these scales, arcsecond arithmetic is the language of the universe's distance ladder.

Two incompatible military systems exist: NATO mil: 6,400 per circle (1 mil = 0.05625° = 3.375′). Used by the US, UK, Germany, and most NATO members. Soviet/Russian mil: 6,000 per circle (1 mil = 0.06° = 3.6′). Still used by Russia, China, and former Soviet bloc countries. The difference is ~6.7% — significant for artillery fire-control. At 5,000 m range, a 10-mil deflection error means 56.25 m (NATO) vs 60 m (Soviet) — a 3.75 m discrepancy that matters in precision engagements. Always confirm which mil system a ballistic table uses before application.

In rotational mechanics, angular velocity ω is in radians per second (rad/s) — because the tangential velocity formula v = rω only works without a conversion factor in radians. A motor at 3,000 RPM = 3000 × 2π/60 = 314.16 rad/s. Angular acceleration α is in rad/s². Kinetic energy of rotation: KE = ½Iω² (I in kg·m², ω in rad/s). In wave physics, phase and angular frequency ω = 2πf are always in radians — so a 50 Hz power supply has ω = 314.16 rad/s. Every periodic phenomenon in physics is naturally expressed in radians because the exponential form e^{iωt} requires the angle to be dimensionless — which radians are, by definition.

MOA (Minute of Angle) = 1/60° = 1 arcminute. At 100 yards, 1 MOA ≈ 1.047 inches (often approximated as 1 inch). At 100 m, 1 MOA ≈ 2.908 cm. MOA is preferred in US rifle shooting because scope adjustments at 100-yard benchmarks correspond to near-integer inch values. Mil (NATO) at 100 m = 10 cm exactly — preferred for metric ranges and military applications. Converting: 1 mil = 3.438 MOA; 1 MOA = 0.291 mil. Both measure the same angular spread — the choice is cultural and mathematical convenience. Long-range shooters often memorise both systems.

Industrial robots express joint positions in degrees (displayed) and radians (internally computed). A 6-axis robot arm stores orientation as rotation matrices or quaternions — all radian-based. CNC rotary axes (A, B, C) move in degrees with resolution as fine as 0.001°. Stepper motors: a standard motor has 200 steps/revolution = 1.8° per step; with 1/16 microstepping, resolution = 0.1125° per step. Servo encoders report in encoder counts (e.g., 10,000 counts/rev), requiring counts × 360/10000 = degrees conversion. Angular precision in robotics is about repeatable positioning — even ±0.01° cumulates to millimetre-level end-effector error in a 1 m arm.

The 360-degree circle is Babylonian in origin, approximately 2000–1500 BCE. The Babylonians used a base-60 (sexagesimal) number system and observed that the Sun moves roughly 1° per day along the ecliptic (completing 360° in one year — though the solar year is actually 365.25 days, the Babylonian calendar used 360-day approximations). 360 was also chosen for its exceptional divisibility: it has 24 divisors — including 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360 — making equal subdivision into halves, thirds, quarters, sixths, eighths, and twelfths all yield whole numbers. This property made construction, navigation, and calendar division far easier with integer arithmetic.