Quick Reference — Common Angles
Enter a valid angle value.
Decimal places:
Converted Angle
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All Angle Units
Degrees – Minutes – Seconds (DMS)
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Format: D° M' S" — used in GPS, astronomy, navigation, and surveying
⚠️ Disclaimer: Results use full double-precision floating-point arithmetic. For extremely precise scientific or engineering work, verify against authoritative standards. Milliradian values here use pure math milliradians (1 mrad = 0.001 rad), not NATO mils (6400/circle) or Warsaw Pact mils (6000/circle).
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Sources & Methodology
All conversion formulas verified mathematically. Base unit = degrees. Every unit has a known degrees-equivalent, and all conversions pass through degrees as the common base. Formulas cross-referenced against ISO 80000-3 (quantities and units for mechanics, including angle).
BIPM — Bureau International des Poids et Mesures
International authority on measurement units. Reference for radian definition (SI unit of angle), degree relationship to radian (1 revolution = 2*pi radians = 360 degrees), and gradian definition used in this calculator.
NIST — SI Units of Angle
National Institute of Standards and Technology documentation on angle measurement units, radian as the SI derived unit, and practical conversions between degrees, radians, and gradians used in science and engineering.
Verified Conversion Method:
target = input x (source_to_degrees / target_to_degrees)
Degrees per unit: degrees=1 | radians=180/pi | gradians=0.9 | turns=360 | arcmin=1/60 | arcsec=1/3600 | mrad=180/(pi x 1000)
DMS: degrees=floor(decimal) | minutes=floor((decimal-degrees) x 60) | seconds=((decimal-degrees) x 60 - minutes) x 60
Test: 90deg = pi/2 rad = 100 grad = 0.25 turns = 5400 arcmin = 324000 arcsec = 1570.796 mrad. All verified correct.
Angle Unit Conversion — Complete Guide 2026
Angles can be measured in seven different units, each designed for a specific context. Degrees dominate everyday use, engineering drawings, and navigation. Radians are the standard in mathematics, physics, and programming. Gradians simplify surveying calculations. Arcminutes and arcseconds measure precise astronomical positions and GPS coordinates. Understanding when each unit is used — and how to convert between them — is essential for students, engineers, programmers, and scientists.
The Universal Conversion Method (Degrees as Base)
target = input x (source_to_degrees / target_to_degrees)
Every angle unit has a fixed relationship to degrees. To convert any unit to any other unit, first express the input in degrees, then rescale to the target unit.
Degrees-per-unit values:
1 degree = 1 degree
1 radian = 180/π degrees ≈ 57.2958 degrees
1 gradian = 0.9 degrees (9/10)
1 turn = 360 degrees
1 arcminute = 1/60 degree
1 arcsecond = 1/3600 degree
1 milliradian = 180/(π x 1000) degrees ≈ 0.0572958 degrees
Degrees-per-unit values:
1 degree = 1 degree
1 radian = 180/π degrees ≈ 57.2958 degrees
1 gradian = 0.9 degrees (9/10)
1 turn = 360 degrees
1 arcminute = 1/60 degree
1 arcsecond = 1/3600 degree
1 milliradian = 180/(π x 1000) degrees ≈ 0.0572958 degrees
Common Angle Reference Table
| Degrees | Radians | Gradians | Turns | Arcminutes |
|---|---|---|---|---|
| 0° | 0 | 0 gon | 0 | 0' |
| 30° | π/6 ≈ 0.5236 | 33.33 gon | 0.0833 | 1,800' |
| 45° | π/4 ≈ 0.7854 | 50 gon | 0.125 | 2,700' |
| 60° | π/3 ≈ 1.0472 | 66.67 gon | 0.1667 | 3,600' |
| 90° | π/2 ≈ 1.5708 | 100 gon | 0.25 | 5,400' |
| 180° | π ≈ 3.1416 | 200 gon | 0.5 | 10,800' |
| 270° | 3π/2 ≈ 4.7124 | 300 gon | 0.75 | 16,200' |
| 360° | 2π ≈ 6.2832 | 400 gon | 1.0 | 21,600' |
When to Use Each Angle Unit
- Degrees: Everyday geometry, architecture, navigation compasses, mechanical engineering drawings, meteorology wind direction. Intuitive for most people — a right angle is 90°, a full circle is 360°.
- Radians: Mathematics (calculus, trigonometric series), physics, programming (sin(), cos(), tan() in most languages expect radians), signal processing, control theory. The SI-derived unit for angle.
- Gradians (gon): Surveying and civil engineering in continental Europe, especially France. A right angle = 100 gradians produces clean decimal arithmetic. Scientific calculators have a GRAD mode for this unit.
- Turns (revolutions): Engineering for rotating machinery — motors, screws, gears. A motor specification of 1,500 RPM directly uses turns per minute. Avoids dealing with 360-degree multiples.
- Arcminutes and arcseconds: Astronomy (star positions, telescope resolution), GPS coordinates (latitude and longitude), navigation (1 arcminute ≈ 1 nautical mile at the equator), precision surveying and geodesy.
- Milliradians (mrad): Long-range shooting scopes (reticle adjustments), artillery and mortar range estimation, optics engineering, ballistics. At 1000 metres, 1 mrad ≈ 1 metre of arc — useful for windage and elevation adjustments.
Degrees-Minutes-Seconds (DMS) Format
DMS is the traditional way of writing angles more precisely than a decimal degree allows. It works like time: 1 degree = 60 arcminutes, 1 arcminute = 60 arcseconds. GPS coordinates are frequently displayed in DMS: 51° 30' 26" N means 51 degrees, 30 arcminutes, 26 arcseconds North. To convert decimal degrees to DMS: take the whole number as degrees, multiply the decimal remainder by 60 to get minutes, then multiply that remainder by 60 again to get seconds.
💡 Programmer tip: JavaScript's Math.sin(), Math.cos(), and Math.tan() all expect input in radians. If you have an angle in degrees, convert first:
Math.sin(degrees * Math.PI / 180). Forgetting this conversion is one of the most common bugs in geometry-related code.Frequently Asked Questions
Radians = Degrees x (pi / 180). Approximately: Degrees x 0.017453. Example: 90 degrees x (pi/180) = pi/2 radians = 1.5708 rad. To reverse: Degrees = Radians x (180 / pi). Example: pi/4 radians x (180/pi) = 45 degrees. Radians are required for calculus and most programming trigonometric functions.
Gradians = Degrees x (10/9) = Degrees x 1.11111. A full circle has 400 gradians vs 360 degrees, so the ratio is 400/360 = 10/9. Example: 90 degrees x (10/9) = 100 gradians. A right angle is always 100 gradians. To reverse: Degrees = Gradians x (9/10). Gradians are used in European surveying where decimal right-angle arithmetic is convenient.
Turns = Degrees / 360. A full circle is 1 turn. Example: 90 degrees / 360 = 0.25 turns. 180 degrees = 0.5 turns. To reverse: Degrees = Turns x 360. Turns are used in mechanical engineering for rotating systems like motors, gears, and screws where counting complete rotations is practical.
1 degree = 60 arcminutes (symbol '). 1 arcminute = 60 arcseconds (symbol "). So 1 degree = 3600 arcseconds. Example: 1.5 degrees = 1 degree 30 arcminutes. These units are used in astronomy, GPS coordinates, and navigation. At the equator, 1 arcminute of latitude corresponds to approximately 1.85 km -- the origin of the nautical mile.
Step 1: Whole number = degrees. Step 2: Decimal x 60 = minutes (take whole number). Step 3: Remaining decimal x 60 = seconds. Example: 40.4461 degrees. Degrees = 40. Minutes = 0.4461 x 60 = 26.766, so 26 arcminutes. Seconds = 0.766 x 60 = 45.96 arcseconds. Result: 40 degrees 26' 45.96".
A gradian (gon, grad) divides a full circle into 400 equal parts, making a right angle exactly 100 gradians. Introduced during the French metric reform for its decimal-friendly properties. Used in surveying and civil engineering in continental Europe. Scientific calculators have a GRAD mode. Converting between gradians and degrees: multiply by 9/10 (gon to deg) or 10/9 (deg to gon).
A full circle is exactly 2*pi radians, approximately 6.2832 radians. This comes from the circumference formula: circumference = 2 x pi x radius. At radius 1, circumference = 2*pi, which equals the angle in radians. Key values: 0=0, pi/6=30deg, pi/4=45deg, pi/3=60deg, pi/2=90deg, pi=180deg, 3pi/2=270deg, 2pi=360deg.
Gradians = Radians x (200 / pi). Approximately: Radians x 63.662. Example: pi/2 radians x (200/pi) = 100 gradians. 1 radian x (200/pi) = 63.66 gradians. To reverse: Radians = Gradians x (pi / 200). Approximately Gradians x 0.015708.
A milliradian (mrad) is 0.001 radians. A full circle contains 2000*pi (approximately 6283) milliradians. At 1000 metres, 1 mrad corresponds to approximately 1 metre of arc length, making it useful for long-range shooting scope adjustments, artillery range estimation, and optics engineering. Note: NATO uses 6400 mils/circle and Warsaw Pact uses 6000 mils/circle -- slightly different from pure math mrad.
Radians make calculus simpler. The derivative of sin(x) is cos(x) only when x is in radians. Taylor series sin(x) = x - x^3/6 + x^5/120 and cos(x) = 1 - x^2/2 + x^4/24 are valid only in radians. Arc length = radius x angle (in radians), with no conversion factor. Most programming languages (JavaScript, Python, C++) implement sin(), cos(), tan() expecting radians for this reason.
Degrees = Arcminutes / 60. Example: 90 arcminutes / 60 = 1.5 degrees. For arcseconds to degrees: Degrees = Arcseconds / 3600. Example: 7200 arcseconds / 3600 = 2 degrees. The reverse: 1 degree = 60 arcminutes = 3600 arcseconds. GPS coordinates use decimal degrees or DMS. 1 arcminute at the equator equals approximately 1.85 km (the origin of the nautical mile).
Essential pairs: 0=0, 30deg=pi/6, 45deg=pi/4, 60deg=pi/3, 90deg=pi/2, 120deg=2pi/3, 135deg=3pi/4, 150deg=5pi/6, 180deg=pi, 270deg=3pi/2, 360deg=2pi. Memorise pi/6, pi/4, pi/3, and pi/2 and you can derive the rest. In gradians: 30deg=33.33, 45deg=50, 60deg=66.67, 90deg=100, 180deg=200, 360deg=400.
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