📐 Angle Conversion Calculator
Convert between degrees, radians, gradians, mils, arcminutes, arcseconds, compass points, turns and 14 angle units — with MathJax formulas for arc length, chord length, sector area, angular velocity, DMS conversion, and the small angle approximation
🔄 Angle Unit Converter
🌐 This Angle in All Units
🕐 DMS ↔ Decimal Degree Converter
Convert between Degrees°Minutes′Seconds″ (DMS) and decimal degrees — used for GPS coordinates, astronomy, and geographic data.
📖 How to Use This Angle Converter
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1Enter Your Angle Value
Type any angle — positive, negative, or decimal. Works from arcseconds (tiny astronomical angles) to millions of degrees (cumulative rotations in engineering).
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2Select From and To Units
Choose source and target from 15 units: degrees, radians, gradians, NATO mils, arcminutes, arcseconds, turns, compass points, astrological signs, binary radians, and circle fractions (1/16 through 1/2).
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3Read the Result & Formula
The teal result box shows the converted value and exact conversion factor. "All Units at Once" shows your angle in every supported unit simultaneously.
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4Use Quick Buttons
Presets — °→rad, rad→°, °→grad, °→arcmin, °→mil — set both dropdowns instantly for the most searched angle conversions.
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5DMS ↔ Decimal
Use the DMS converter above for GPS-style coordinates (e.g., 51°30′26″N for London). Entry of degrees, minutes, and seconds outputs decimal degrees, radians, and gradians simultaneously.
📊 Angle Unit Conversion Reference Table
| Unit | Symbol | Full Circle | In Degrees | In Radians | In Gradians |
|---|---|---|---|---|---|
| Degree | ° | 360 | 1° | \(\pi/180\) | 1.1111 grad |
| Radian | rad | 2π ≈ 6.2832 | 180/π ≈ 57.2958° | 1 | 200/π ≈ 63.662 grad |
| Gradian / Gon | grad | 400 | 0.9° | \(\pi/200\) | 1 grad |
| NATO Mil | mil | 6400 | 0.05625° | \(\pi/3200\) | 0.0625 grad |
| Arcminute | ′ | 21,600 | 1/60° | \(\pi/10800\) | 1/54 grad |
| Arcsecond | ″ | 1,296,000 | 1/3600° | \(\pi/648000\) | 1/3240 grad |
| Turn / Revolution | rev | 1 | 360° | 2π | 400 grad |
| Compass Point | pt | 32 | 11.25° | \(\pi/16\) | 12.5 grad |
| Astrological Sign | sign | 12 | 30° | \(\pi/6\) | 33.333 grad |
| Binary Radian | brad | 256 | 1.40625° | \(\pi/128\) | 1.5625 grad |
| Quadrant (1/4) | — | 4 | 90° | \(\pi/2\) | 100 grad |
| Sextant (1/6) | — | 6 | 60° | \(\pi/3\) | 66.667 grad |
📐 Degrees — The History of the 360° Circle
Why are there 360 degrees in a circle? The answer comes from the Babylonians, who used a base-60 (sexagesimal) number system around 2000 BCE. The number 360 has an extraordinary number of divisors — 24 divisors in total, including 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180 — making it extremely convenient for fractions and dividing circles into equal parts without remainders.
A secondary reason: the Babylonian year was approximately 360 days, so the sun appeared to move approximately 1 degree per day along the ecliptic. This astronomical convenience reinforced the 360-degree convention. Greek astronomers, including Hipparchus (around 150 BCE), built on Babylonian tables and established the 360-degree system as standard. Ptolemy used it in the Almagest, and it was never displaced because of its remarkable computational convenience.
🔄 Radians — The Natural Angle Unit
\( \theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180} \qquad \Leftrightarrow \qquad \theta_{\text{deg}} = \theta_{\text{rad}} \times \frac{180}{\pi} \)
\( \pi\,\text{rad} = 180° \qquad 2\pi\,\text{rad} = 360° \qquad \frac{\pi}{2}\,\text{rad} = 90° \qquad \frac{\pi}{4}\,\text{rad} = 45° \)
\( 1\,\text{rad} = \frac{180°}{\pi} \approx 57.2957795°\qquad 1° = \frac{\pi}{180}\,\text{rad} \approx 0.017453293\,\text{rad} \)
The radian was formally introduced in the late 19th century. The term was coined by James Thomson, brother of Lord Kelvin, in 1873. Before then, angles were measured in degrees exclusively, even in mathematical analysis. The defining property of the radian — that arc length equals radius when angle equals 1 radian — makes it the natural choice for any formula involving circular motion, oscillations, or trigonometry in calculus.
\( 0° = 0\,\text{rad} \qquad 30° = \pi/6 \approx 0.5236\,\text{rad} \qquad 45° = \pi/4 \approx 0.7854\,\text{rad} \)
\( 60° = \pi/3 \approx 1.0472\,\text{rad} \qquad 90° = \pi/2 \approx 1.5708\,\text{rad} \qquad 120° = 2\pi/3 \approx 2.0944\,\text{rad} \)
\( 135° = 3\pi/4 \approx 2.3562\,\text{rad} \qquad 150° = 5\pi/6 \approx 2.6180\,\text{rad} \qquad 180° = \pi \approx 3.1416\,\text{rad} \)
\( 270° = 3\pi/2 \approx 4.7124\,\text{rad} \qquad 360° = 2\pi \approx 6.2832\,\text{rad} \)
📏 Arc Length, Chord Length & Sector Area Formulas
\( s = r\theta \qquad \text{(arc length, } \theta \text{ in radians)} \)
\( s = r \cdot \frac{\theta_{\text{deg}} \cdot \pi}{180} \qquad \text{(arc length, } \theta \text{ in degrees)} \)
\( c = 2r\sin\!\left(\frac{\theta}{2}\right) \qquad \text{(chord length, } \theta \text{ in radians)} \)
\( A_{\text{sector}} = \frac{1}{2}r^2\theta = \frac{\theta_{\text{deg}}}{360}\pi r^2 \qquad \text{(sector area)} \)
\( h = r\left(1 - \cos\frac{\theta}{2}\right) \qquad \text{(sagitta / arc height)} \)
Problem: A circular arc has radius 150 mm and subtends 45°. Find arc length, chord length, and sector area.
Convert to radians: \( \theta = 45 \times \pi/180 = \pi/4 \approx 0.7854\,\text{rad} \)
Arc length: \( s = 150 \times 0.7854 = \mathbf{117.81\,\text{mm}} \)
Chord length: \( c = 2 \times 150 \times \sin(0.7854/2) = 300 \times \sin(22.5°) = 300 \times 0.3827 = \mathbf{114.80\,\text{mm}} \)
Sector area: \( A = \tfrac{1}{2} \times 150^2 \times 0.7854 = \tfrac{1}{2} \times 22500 \times 0.7854 = \mathbf{8835.7\,\text{mm}^2} \)
Sagitta: \( h = 150(1-\cos 22.5°) = 150(1-0.9239) = 150 \times 0.0761 = \mathbf{11.42\,\text{mm}} \)
⚙️ Angular Velocity & Rotational Motion
\( \omega = \frac{\theta}{t} \qquad \text{(angular velocity = angle/time, rad/s)} \)
\( \omega_{\text{rad/s}} = n_{\text{rpm}} \times \frac{2\pi}{60} = n_{\text{rpm}} \times 0.10472 \)
\( n_{\text{rpm}} = \omega_{\text{rad/s}} \times \frac{60}{2\pi} = \omega \times 9.5493 \)
\( v = \omega r \qquad \text{(linear velocity at radius } r \text{)} \)
\( a_c = \omega^2 r = \frac{v^2}{r} \qquad \text{(centripetal acceleration)} \)
🔭 The Small Angle Approximation — Optics, Astronomy & Ballistics
\( \sin\theta \approx \tan\theta \approx \theta \qquad \text{(}\theta\text{ in radians, for } |\theta| \ll 1\text{ rad)} \)
\( \cos\theta \approx 1 - \frac{\theta^2}{2} \qquad \text{(second-order approximation)} \)
\( \text{Error of sin}\approx\theta: \; \varepsilon = \left|\frac{\theta - \sin\theta}{\sin\theta}\right| \approx \frac{\theta^2}{6} \)
\( 1° = 0.01745\,\text{rad}: \; \varepsilon \approx (0.01745)^2/6 \approx 0.005\% \qquad 10° = 0.1745\,\text{rad}: \; \varepsilon \approx 0.506\% \)
📊 Gradians, DMS, Mils & Other Specialised Systems
Gradian (Gon) — Metric Surveying
Divides the full circle into 400 equal parts. One right angle = exactly 100 grad, making field calculations and slope calculations trivially decimal. \(1\,\text{grad} = 0.9° = \pi/200\,\text{rad}\). Adopted in France during the metric revolution (1790s) alongside the metre and kilogram. Still used by surveyors, civil engineers, and geodesists across Europe (especially Switzerland, Germany, the Netherlands). Theodolites in these countries typically display angles in grad.
NATO Mil — Military Precision
Divides the circle into 6400 mils (NATO standard). \(1\,\text{mil} = 0.05625° = \pi/3200\,\text{rad}\). The name "mil" comes from "milliradian" — at 1000 m range, 1 mil corresponds to 1 m lateral displacement (approximately), making field estimation trivial without a calculator: \(\text{object width} = \text{angular width in mils} \times \text{range}/1000\). Used for artillery elevation, mortar adjustment, and binocular range estimation. Note: Soviet/Russian mil = 6000 parts; US WW2 mil = 6400 (current NATO).
DMS — Degrees Minutes Seconds
Subdivides degrees: \(1° = 60'\) (arcminutes); \(1' = 60''\) (arcseconds). GPS coordinates use decimal degrees, but maps, astronomy, and surveying still use DMS. London: 51°30′26″N, 0°7′39″W. The Hubble Space Telescope can resolve objects separated by 0.05 arcseconds. Formula: \(\theta_{\text{dec}} = D + M/60 + S/3600\). The parsec (astronomical distance unit) is defined as the distance at which 1 AU subtends an angle of exactly 1 arcsecond.
Binary Radian (Brad)
Divides the full circle into 256 binary radians (2⁸). \(1\,\text{brad} = 360°/256 = 1.40625°\). Used in computer graphics, DSP (digital signal processing), and embedded systems where integer arithmetic is preferred. A 360° rotation in computer graphics maps to 0–255 (unsigned 8-bit) cleanly. Also used in CNC machines with 8-bit angular resolution. Some radar systems use binary angular units for efficient bit manipulation.
\( \theta_{\text{dec}} = D + \frac{M}{60} + \frac{S}{3600} \qquad \text{(DMS → decimal)} \)
\( D = \lfloor\theta_{\text{dec}}\rfloor \qquad M = \lfloor(\theta_{\text{dec}}-D)\times 60\rfloor \qquad S = ((\theta_{\text{dec}}-D)\times 60 - M)\times 60 \qquad \text{(decimal → DMS)} \)
\( \text{Example: } 51°30'26'' = 51 + 30/60 + 26/3600 = 51 + 0.5 + 0.00722... = 51.50722°\)
\( 1'' \text{ of arc on Earth's surface} \approx 30.9\,\text{m} \qquad 1' \text{ of arc} \approx 1.855\,\text{km} \qquad \text{(definition of nautical mile: }1' = 1\text{ NM)} \)
🛠️ Real-World Applications of Angle Conversion
GPS & Geographic Coordinates
GPS systems store coordinates in decimal degrees internally (double-precision float). Human-readable GPS shows DMS: 51°30′26″N, 0°7′39″W. Mapping software (Google Maps, OpenStreetMap) accepts both. Column 'lat' in GeoJSON files uses decimal degrees. Converting between DMS and decimal is a daily task for GIS analysts, surveyors, and cartographers.
Astronomy & Telescopes
Stellar coordinates (Right Ascension, Declination) use hours-minutes-seconds (RA) and degrees-arcminutes-arcseconds (Dec). The Hubble Space Telescope resolves 0.05″. A 1-arcsecond parallax angle defines 1 parsec = 3.086×10¹⁶ m = 3.26 light-years. Angular diameter of the Sun: 31.6′–32.7′ depending on Earth-Sun distance. Angular diameter of the Moon: 29.4′–33.5′ — nearly identical to Sun's, enabling total solar eclipses.
CNC Machining & CAD
CNC G-code specifies arc moves using I, J, K offsets (Cartesian) or R (radius) with angle. Tool path planners use radians internally but G-code users think in degrees. CAD packages (AutoCAD, SolidWorks) display degrees but pass radians to solvers. Involute gear tooth profile uses \(\theta = \tan\phi - \phi\) (involute function, phi in radians). Cam profiles are computed in radians for smooth velocity profiles.
Game Development & Computer Graphics
3D game engines (Unity, Unreal Engine, Three.js) use radians in all math libraries. Artists set angles in degrees; engines convert internally via \(\theta_\text{rad} = \theta_\text{deg} \times \pi/180\). Rotation matrices use \(\cos\theta\) and \(\sin\theta\) where \(\theta\) is in radians. Quaternion rotation in 3D: half-angle in radians. Binary radians (brads) are used in some older game platforms for efficient integer arithmetic.
❓ Frequently Asked Questions — Angle Conversion
How do I convert degrees to radians?
How do I convert radians to degrees?
What is the formula for converting degrees to gradians?
How do I convert DMS (degrees minutes seconds) to decimal degrees?
How do I convert an angle to arc length in mm?
Why are radians preferred in physics and calculus?
Math.sin(degrees * Math.PI / 180).