An angle measures the amount of planar rotation or inclination between two intersecting lines or rays sharing a common vertex. Across land surveying, architectural CAD modeling, mechanical engineering, satellite navigation (GPS), astronomical observations, and calculus, angular measurement relies on four primary global standards: sexagesimal circle divisions (Degrees / °, Arcminutes / ‘, Arcseconds / “), SI metric pure ratios (Radians / rad), French metric decimal surveying units (Gradians / Grads / Gons / ^g), and military artillery directional units (NATO Angular Mils, Revolutions / Turns, Quadrants / Right Angles, Sextants).
Our free online Angle Converter provides instant, high-precision conversions across all trigonometric, surveying, astronomical, and military angle units:
- Radians to Degrees (rad to °): Multiply radians by
180 ÷ π(1 rad ≈ 57.295779513° = 3,437.75′ = 206,264.81″ = 63.662 gon). - Degrees to Radians (° to rad): Multiply degrees by
π ÷ 180(1° ≈ 0.0174532925 rad = 60′ = 3,600″ = 1.11111 gon). - Gradians (Gons) to Degrees: Multiply grads by
0.9(400 gon = 360° = 2π rad = 1 full turn). - Arcminutes and Arcseconds:
1 Degree = 60 Arcminutes (') = 3,600 Arcseconds ("). - NATO Mils to Degrees:
1 NATO Mil = 0.05625° = 3.375'(6,400 NATO mils = 360° = 2π rad). - Full Revolution / Turn / Circle:
1 Turn = 360° = 2π rad ≈ 6.2831853 rad = 400 gon = 4 Quadrants.
Master Angle & Rotation Unit Conversion Table
The table below displays exact mathematical relationships, SI radian multipliers, and sexagesimal equivalents relative to 1 Degree (° = 1/360 of a full circle):
| Angle Unit Name | Symbol | Exact Value in Degrees (°) | Radian (rad) & Grad (gon) Equivalent | Domain & System Standard |
|---|---|---|---|---|
| 1 Degree | ° |
1.0° (1/360 turn = 60′ = 3,600″) |
0.01745329 rad5 (π/180 rad / 1.11111 gon) |
Universal Planar Geometry & Navigation Standard |
| 1 Radian | rad |
57.295779513° (180/π deg) |
1.0 rad (63.661977 gon / 3,437.75′) |
SI Derived Dimensionless Unit (Arc Length s = r·θ) |
| 1 Grad / Gradian / Gon | ^g, gon |
0.90° (9/10 deg = 54′ = 3,240″) |
0.01570796 rad (π/200 rad / 1.0 gon) |
French Metric Geodetic Surveying Standard |
| 1 Arcminute | ', MOA |
0.016666667° (1/60 deg = 60″) | 0.000290888 rad (0.0185185 gon) | Ballistics Sight Adjustment & Vision Optics |
| 1 Arcsecond | " |
0.000277778° (1/3,600 deg = 1/60′) | 0.000004848 rad (0.0003086 gon) | Astronomical Telescope Resolution & GIS Mapping |
| 1 NATO Angular Mil | mil |
0.056250° (360/6,400 deg = 3.375′) |
0.000981748 rad (π/3,200 rad / 0.0625 gon) | NATO Military Artillery & Compass Target Azimuth |
| 1 Sign (Zodiac) | sign |
30.0° (1/12 full circle) |
0.52359877 rad (π/6 rad / 33.333 gon) | Historical Ecliptic Astronomical Constellation Division |
| 1 Sextant | sextant |
60.0° (1/6 full circle) |
1.04719755 rad (π/3 rad / 66.667 gon) | Maritime Celestial Navigation Arc Instrument |
| 1 Quadrant / Right Angle | quadrant |
90.0° (1/4 full turn) |
1.57079633 rad (π/2 rad / 100.0 gon) |
Perpendicular Geometry & Cartesian Quadrant |
| 1 Revolution / Turn / Circle | r, turn |
360.0° (Full Rotation) |
6.28318531 rad (2π rad / 400.0 gon / 6,400 mils) |
Complete Circular Rotation Metric |
Step-by-Step Trigonometric Calculation Example
To convert an angle of 2.5 Radians (2.5 rad) into Degrees (°) and Gradians (gon):
2.5 rad × (180 ÷ π) = 2.5 × 57.295779513 = 143.2394 Degrees (143.24°)
2.5 rad × (200 ÷ π) = 2.5 × 63.661977237 = 159.1549 Grads (159.15 gon)
Thus, 2.5 radians equals 143.24 Degrees (or 159.15 Grads / 143° 14′ 22″ in sexagesimal DMS format).
Real-World Surveying, Astronomy & Optical Benchmarks
Below is a comparative reference chart showing angular measurements across human vision, GPS surface mapping, telescope optics, and circle divisions:
| Angular Physical Context / Subject | Angle in Sexagesimal Units (° / ‘ / “) | Angle in Radians & Metric Units (rad / gon / mil) | Physical & Engineering Context |
|---|---|---|---|
| Hubble Space Telescope Angular Resolution | 0.05 Arcseconds (0.05″) | 2.424 × 10-7 rad (0.0000154 gon) | Diffraction-limited optical resolving power in space |
| Human Eye Visual Acuity Resolving Limit | 1.0 Arcminute (1.0′ = 60″) | 0.0002909 rad (0.01852 gon) | Snellen 20/20 vision angular resolution threshold |
| Earth Surface Latitude Distance for 1 Arcsecond | 1.0 Arcsecond (1.0″ = 0.0002778°) | 4.848 × 10-6 rad (≈ 30.86 meters / 101.2 ft) | Equatorial arc length per arcsecond of latitude on Earth |
| Sun and Full Moon Angular Diameter from Earth | 0.50° to 0.53° (30′ to 32′) | 0.0087 to 0.0093 rad (0.555 to 0.589 gon) | Apparent sky disk size enabling total solar eclipses |
| Military NATO Compass Azimuth Full Circle | 360.0° (1 Full Turn) | 6,400 NATO Mils (2π rad / 400.0 gon) | Standard NATO artillery and tactical compass circle grid |
History & Standards: 2000 BC Babylonian 360° Circle vs. 1795 French Gradian vs. 1873 Radian Formalization
The 360-Degree Sexagesimal Circle (c. 2000 BC)
The division of a circle into 360 degrees dates back over 4,000 years to ancient Sumerian and Babylonian astronomers in Mesopotamia. Babylonian mathematicians used a base-60 (sexagesimal) numbering system. Because their ideal calendar estimated the solar year at 360 days, they divided the sun’s annual path across the ecliptic into 360 equal daily steps. Furthermore, stepping off a circle’s radius along its perimeter divides it naturally into 6 equilateral triangles (6 × 60° = 360°).
The 1795 French Decimal Gradian (Gon)
During the French Revolution in 1795, the French National Assembly attempted to decimalize all physical units under the metric system. Engineers established the Gradian (Grad / Gon), defining a right angle as exactly 100 grads (making a full circle 400 grads). Under this system, 1 centigrad of latitude arc along a meridian equaled exactly 1 kilometer. While widely adopted in European land surveying and geodetic civil engineering, the traditional 360° circle remained global standard for navigation and astronomy.
1873 James Thomson Radian Formalization
In 1873, Scottish engineer James Thomson (brother of Lord Kelvin) formally coined the term radian in examination questions at Queen’s College Belfast, later publishing the concept in 1883. Radians represent pure dimensionless ratios where arc length equals radius (s = r · θ), rendering calculus derivative equations (such as d/dx(sin x) = cos x) elegant and free of arbitrary π/180 conversion factors.
Popular direct conversions:
Frequently Asked Questions (FAQ)
How many Degrees (°) are in 1 Radian (rad)?
There are approximately 57.2957795 degrees (180 ÷ π) in 1 Radian.
How many Radians (rad) are in 1 Full Circle (360°)?
There are exactly 2π Radians (approximately 6.2831853 rad) in 1 full circle (360 degrees).
What is a Gradian (Gon)?
A Gradian (Grad or Gon) is a metric unit of angle equal to 0.9 degrees. A right angle equals exactly 100 grads, and a full circle equals 400 grads.
What is a NATO Angular Mil?
A NATO Angular Mil is a military directional unit equal to 1 ÷ 6,400 of a circle (0.05625 degrees). At a range of 1,000 meters, 1 mil subtends an arc of approximately 1 meter, making it ideal for artillery aiming.