Angle Converter

One direction, seven angle units

An angle converter changes the numerical unit used to describe a rotation; it does not rotate the ray or alter the geometry. This tool first converts the source value to turns. Every output is calculated directly from that unrounded value instead of converting a rounded display again, which avoids cumulative drift when radians, arcseconds, or several other formats appear in one task.

Enter a source value, identify its unit, choose a target, and set display precision. Keep raw preserves negative angles and rotations beyond one turn. The normalization choices map the same direction to 0°–360° or −180°–180°. The angle converter result, π notation, equivalent-unit list, and live ray all come from the same calculation value.

How to convert angles without rounding drift

Enter a finite value and identify its source unit. Do not assume an unlabeled number is degrees when it came from code, navigation notes, or a calculator set to radians.
Choose the target unit. Swap reverses the unit fields using the calculated value rather than a shortened display label.
Choose Keep raw for signed rotations and multiple turns, 0–360° for an unsigned direction, or −180°–180° for a signed direction convention.
Set display precision after choosing the convention. Precision changes visible rounding only; the angle converter keeps the independent turns value.
Use a preset to compare a common degree angle with decimal radians and a fraction of π, then copy the result or share the numeric state.

Conversion formulas and unit definitions

Turns are the common model: turns = degrees ÷ 360. Radians = turns × 2π, gradians = turns × 400, and degrees = turns × 360.
One degree contains 60 arcminutes, and one arcminute contains 60 arcseconds. Therefore arcminutes = degrees × 60 and arcseconds = degrees × 3,600.
This page uses NATO mil (6400 per turn), so NATO mils = turns × 6,400. It never substitutes an ambiguous plain mil or a different artillery circle.
A π fraction appears only when its coefficient is within a strict tolerance of a reduced fraction with a limited denominator. Other values remain decimal multiples of π.
Scientific notation is reserved for values too large or too small for the selected fixed precision; ordinary results stay in decimal notation.

Worked angle conversion examples

Degrees to radians: 180° ÷ 360 = 0.5 turn, and 0.5 × 2π = π radians. Its ray remains a half turn from zero.
Radians to degrees: π/2 radians is one quarter turn, so radians to degrees gives 90°. The decimal radian value is approximately 1.570796.
Fine units: 1° equals 60 arcminutes and 3,600 arcseconds. Conversely, 30 arcminutes equals 0.5°.
NATO division: one full turn equals 6,400 NATO mils; 1,600 NATO mils is 90°.
Normalization: 450° remains 450° in Keep raw and becomes 90° in either directional range; −90° becomes 270° in the unsigned range.

Why one turn has several equivalent values

Degrees divide a rotation into 360 parts, gradians or gons divide it into 400, and turns use the complete rotation itself. Radian measure relates arc length to radius, so circumference 2πr makes a full rotation 2π radians. Arcminutes and arcseconds subdivide degrees, while the defined NATO circle has 6,400 mils.

The equality 360° = 2π rad = 400 grad = 1 turn = 21,600 arcminutes = 1,296,000 arcseconds = 6,400 NATO mils describes one rotation. Converting angles changes its scale label, not the final geometric direction.

π fractions and normalization boundaries

Common angles produce familiar fractions: 30° is π/6, 45° is π/4, 60° is π/3, and 90° is π/2. The fraction matcher is conservative so a nearby measured decimal is not declared exact.

Unsigned normalization uses [0°, 360°), so 360° maps to 0°. Signed normalization uses [−180°, 180°), so the positive boundary is represented as −180°. Keep raw when rotation count or original sign matters.

What a unit conversion does not establish

The angle converter performs deterministic unit arithmetic. It cannot improve a measured angle, decide whether clockwise or counterclockwise was intended, identify the north reference for a bearing, or recover a unit omitted from source material. Extra displayed decimal places are not evidence of extra measurement accuracy.

All calculations run in the browser. A share URL stores only the entered number, source and target units, normalization, and precision; normal use sends no angle input to a server.

Frequently Asked Questions

Why does this angle converter use turns internally?

Every supported unit has a direct relationship to a full rotation. Converting the original source once creates a stable value from which every output can be calculated without chaining rounded displays.

Are a turn and a revolution different here?

No. Both mean one complete 360° rotation; the combined label prevents confusion with a direction command.

Why did 180° normalize to −180°?

The signed range includes −180° and excludes +180°. Both point the same way, and using one boundary avoids duplicate representations.

Does NATO mil mean milliradian?

Not exactly. This page explicitly uses NATO mil (6400 per turn). A mathematical milliradian is not silently substituted.

Why is a radian result a decimal multiple of π?

Only values within a strict tolerance of a simple fraction receive notation such as π/6. A nonmatching value remains decimal so the display does not invent exactness.