About

Miscalculating angle conversions propagates errors through every downstream computation in navigation, surveying, ballistics, and structural engineering. A single wrong conversion between rad and deg can rotate a CNC tool path by 57.3°, destroy a workpiece, or misalign a satellite dish. This calculator converts between 12 angle measurement systems using exact mathematical factors referenced to the degree as the base unit. It handles edge cases including negative angles, values exceeding full rotation, and very small angles used in precision optics.

The tool covers standard units (degrees, radians, gradians) alongside specialized units used in military rangefinding (mrad, NATO mils), precision shooting (MOA, arcseconds), astronomy (hour angles), and legacy computing (binary degrees). All conversions are bidirectional and computed in real time. Note: this tool does not normalize angles to a canonical range. An input of 720° converts directly without reducing to 360°, preserving multi-revolution values required in robotics and winding calculations.

Formulas

All conversions route through degrees as the canonical base unit. The general two-step formula is:

result = input × F_sourceF_target

where F_source is the factor that converts one unit of the source to degrees, and F_target is the factor that converts one unit of the target to degrees.

For the radian-degree relationship specifically:

deg = rad × 180π

For gradians:

deg = gon × 0.9

For milliradians:

deg = mrad × 1801000 × π

Variable legend: F = conversion factor (degrees per one unit). input = numeric value entered by user. result = converted output value. π ≈ 3.14159265358979.

Reference Data

Unit	Symbol	Per Full Circle	Equivalent in Degrees	Primary Domain
Degree	°	360	1°	General, navigation, education
Radian	rad	2π	57.29578°	Mathematics, physics, engineering
Gradian (Gon)	gon	400	0.9°	Surveying, civil engineering (Europe)
Turn (Revolution)	tr	1	360°	Rotational mechanics, turbines
Milliradian	mrad	6283.185	0.05730°	Military rangefinding, ballistics
Minute of Arc (MOA)	′	21600	0.01667°	Precision shooting, optics
Arcsecond	″	1296000	0.000278°	Astronomy, geodesy
NATO Mil	mil	6400	0.05625°	NATO military artillery
Binary Degree	brad	256	1.40625°	8-bit game engines, embedded systems
Quadrant	quad	4	90°	Classical geometry, navigation
Sextant	sxt	6	60°	Celestial navigation instrument
Hour Angle	ha	24	15°	Astronomy, right ascension
Compass Point	cp	32	11.25°	Marine/aviation navigation

Frequently Asked Questions

Floating-point arithmetic in IEEE 754 double-precision cannot represent π exactly. The value 3.141592653589793 is the closest 64-bit representation of π, accurate to 15 significant digits. For symbolic exactness, treat the result as π when the input is exactly 180°. This tool preserves full double-precision fidelity without rounding.

A true milliradian is 1/1000 of a radian, yielding approximately 6283.185 per full circle. A NATO mil is a military approximation that divides the circle into exactly 6400 equal parts for simpler field calculations. One NATO mil equals 0.05625°, while one true mrad equals 0.05730°. The difference is roughly 1.85%, which matters at long-range ballistic distances beyond 1000m.

Temperature does not affect the mathematical conversion between angle units. However, temperature gradients cause atmospheric refraction that bends light paths, introducing apparent angular errors in surveying and astronomy. At sea level, refraction can shift a celestial object's apparent position by up to 34′ (arcminutes) near the horizon. This tool computes pure mathematical conversions and does not model atmospheric corrections.

Gradians (gon) are standard in European surveying and civil engineering because a right angle equals exactly 100gon, making percentage-grade calculations trivial. A 1% slope equals 1gon for small angles. Most total stations manufactured by Leica and Trimble default to gradians. Use degrees for navigation, radians for calculus and physics, and gradians for land surveying when working with European standards or DIN specifications.

Yes. The converter applies the conversion factor linearly without clamping or normalizing. An input of 720° converts to 4π rad (approximately 12.566). Negative values are preserved: −90° becomes −100gon. This behavior is intentional for applications like robotics joint rotation tracking and helical path computation where cumulative angle matters.

A binary degree (brad) divides the full circle into 256 parts, fitting neatly into an unsigned 8-bit integer (0 - 255). This was common in early video game engines (Doom, Quake) and remains used in embedded microcontrollers with limited memory. One brad equals 1.40625°. The 16-bit variant uses 65536 divisions and appears in digital compasses and IMU firmware.