About

Angular measurement errors propagate through every downstream calculation in surveying, navigation, CNC machining, and structural engineering. A 0.1° misalignment over a 100m span produces a lateral deviation of roughly 17.5cm. This converter handles 12 angular units with full precision, using rad as the canonical intermediary. It covers SI radians, ISO degree subdivisions (arcminutes, arcseconds), NATO milliradians (6400 per revolution), gradians used in continental European surveying, and astronomical hour angles. All conversions are exact algebraic ratios involving π. No floating-point rounding is applied to conversion factors themselves.

Limitations: this tool assumes a flat Euclidean plane. For geodetic work on ellipsoidal surfaces, raw angular conversions must be supplemented with datum-specific corrections. Precision is bounded by IEEE 754 double-precision (≈15 - 17 significant digits). Pro tip: when converting arcseconds for astronomical right ascension, remember that 1 hour angle = 15°, not 1°.

Formulas

All conversions route through radians as the canonical base unit. For any source unit with value v, the conversion to a target unit follows a two-step process:

v_target = v_source × k_source→radk_target→rad

The core degree-radian relationship derives from the definition of a full circle:

360° = 2π rad ∴ 1° = π180 rad

Where: v_source = numeric input value. k_source→rad = conversion factor from the source unit to radians. k_target→rad = conversion factor from the target unit to radians. Since both factors are constants, the result simplifies to a single multiplication by the ratio of factors. No intermediate rounding is performed.

Reference Data

Unit	Symbol	Per Full Revolution	Equivalent in Degrees	To Radians Factor	Common Domain
Degree	°	360	1°	π ÷ 180	General, Navigation
Radian	rad	2π	≈ 57.2958°	1	Mathematics, Physics
Gradian (Gon)	grad	400	0.9°	π ÷ 200	Surveying (Europe)
Turn (Revolution)	tr	1	360°	2π	Engineering, Rotary
Arcminute	′	21,600	1/60°	π ÷ 10,800	Astronomy, Optics
Arcsecond	″	1,296,000	1/3600°	π ÷ 648,000	Astronomy, Geodesy
Milliradian (NATO)	mil	6,400	0.05625°	2π ÷ 6,400	Military, Ballistics
Milliradian (SI)	mrad	≈ 6,283.19	≈ 0.0573°	0.001	Optics, Laser Alignment
Sextant	sxt	6	60°	π ÷ 3	Historical Geometry
Quadrant	quad	4	90°	π ÷ 2	Cartography
Hour Angle	ha	24	15°	π ÷ 12	Astronomy (RA)
Compass Point	pt	32	11.25°	π ÷ 16	Marine Navigation
Binary Degree	brad	256	1.40625°	π ÷ 128	Digital Sensors, Games
Sign (Zodiac)	sign	12	30°	π ÷ 6	Traditional Astronomy

Frequently Asked Questions

Gradians divide a right angle into exactly 100 units, making percentage-of-slope calculations trivial. A 1% grade equals 1 gradian. This eliminates the need for trigonometric conversion in field work. Continental European surveying instruments (theodolites, total stations) are calibrated in gradians by default per ISO 17123.

An SI milliradian is exactly 0.001 radians, yielding approximately 6,283.19 per full revolution. NATO milliradians round this to exactly 6,400 per revolution for simpler mental arithmetic in artillery calculations. The NATO mil is therefore about 1.85% larger than the SI mrad. Swedish and Russian military systems use 6,000 and 6,000 divisions respectively, so always confirm the convention before converting.

IEEE 754 double-precision provides roughly 15-17 significant decimal digits. For most engineering work this is more than sufficient. However, repeated conversions (e.g., degrees → radians → degrees → radians) accumulate rounding errors. A single round-trip conversion of 1° introduces an error on the order of 10⁻¹⁶ radians. For sub-arcsecond geodetic work, use arbitrary-precision libraries or maintain values in their native unit until the final computation.

Arcseconds are standard in astronomy (star catalogs, telescope pointing) and geodesy (latitude/longitude). One arcsecond of latitude equals approximately 30.87 meters on Earth's surface. Milliradians are preferred in optics and ballistics because the mrad scale is linear at practical distances: 1 mrad subtends exactly 1 meter at 1,000 meters range, simplifying range estimation without lookup tables.

Binary degrees (brads) divide the circle into 256 parts, fitting exactly into one unsigned 8-bit byte. This makes modular arithmetic trivial: overflow at 256 automatically wraps the angle past 360°. Game engines, MEMS gyroscopes, and embedded compass modules frequently use 8-bit or 16-bit (65,536 divisions) binary angle measurement (BAM) for speed and memory efficiency. One brad equals 1.40625°.

Yes. The converter processes any real number without clamping. Negative values represent clockwise rotation in the standard mathematical convention. Values exceeding one full revolution (e.g., 720°) are converted directly - no automatic normalization is applied, preserving the original magnitude for applications like cumulative rotation tracking in robotics or multi-turn potentiometers.