Angle Converter Calculator
An angle can describe a rotation in a drawing, the phase of a wave, the slope of a roof, the sweep of a CNC tool path, or the argument passed into a trigonometric function. This calculator converts that angle among four circular units: degrees, radians, gradians, and turns. It is a math-angle tool, not a latitude-longitude tool. Geographic coordinates also use degrees, but a coordinate needs extra context such as north or south latitude, east or west longitude, and sometimes a datum or projection.
The converter has one numeric value plus a From Unit and To Unit selector. The default example is 180 degrees to radians. The result panel always labels the selected source and target pair. The radians-to-degrees preset starts with pi radians and keeps the same pair after a refresh; its value and unit fields can also be supplied in the URL. Reverse the selectors to convert degrees back to radians.
For the radians-to-degrees pair, the result details preserve the entered radians, the equivalent fraction of a turn, and the degrees-per-radian factor. The inverse pair likewise preserves the entered degrees, turns, and the pi-radians-to-180-degrees relationship. Negative and multi-turn angles are converted as entered rather than wrapped. Calculations retain full finite numeric precision and round only the displayed text, so copyable output stays consistent with the visible result while the unrounded value remains available.
For other pairs, the result panel lists the remaining supported units except the input and output units. That detail matters when checking work: if you convert gradians to turns, you will also see degrees and radians for the same rotation.
For DMS subdivisions of a degree, use the degrees minutes seconds calculator, degrees to minutes converter, or degrees to seconds converter. For common trigonometry-specific conversions, the degrees to radians converter and radians to degrees converter focus on the most common pair.
What the calculation does
The calculator defines a conversion factor for each unit as the number of radians represented by one unit. One degree is stored as pi divided by 180 radians, one radian is stored as 1, one gradian is stored as pi divided by 200 radians, and one turn is stored as 2 pi radians. When you submit a value, the calculation checks that the value is finite and that both unit names exist in that factor table.
The conversion precomputes the ratio of the input unit’s radian factor to the target unit’s radian factor, then multiplies the input by that ratio. This avoids creating and rounding an intermediate radian value. The primary result is normally formatted with up to six decimal places and a unit symbol: °, rad, grad, or turns. The degrees-to-radians pair uses up to nine decimal places. Alternate result items use the same factor-ratio method, with the current input and target units filtered out. The copy text preserves the entered number, input label, converted number, and target label.
Formula
The general conversion used by the converter is:
The four factors are:
A worked conversion
For a worked example, convert 180 degrees to radians. The input factor for degrees is pi divided by 180. The target factor for radians is 1.
For this pair, the result is displayed with up to nine decimal places: 3.141592654 rad. The details show the entered 180 deg, 0.5 turns, and the relationship pi rad = 180 deg.
Reference table
| Rotation | Degrees | Radians | Gradians | Turns |
|---|---|---|---|---|
| Right angle | 90 | pi divided by 2 | 100 | 0.25 |
| Straight angle | 180 | pi | 200 | 0.5 |
| Three-quarter turn | 270 | 3 pi divided by 2 | 300 | 0.75 |
| Full rotation | 360 | 2 pi | 400 | 1 |
| Double rotation | 720 | 4 pi | 800 | 2 |
Where each unit appears
Degrees are familiar in navigation, protractors, CAD drawings, construction plans, map bearings, and classroom geometry. Radians dominate calculus, physics, signal processing, and most programming language trigonometry functions because derivatives and arc-length formulas are simplest in radians. Gradians, also called gons in some references, divide a right angle into 100 parts and show up in some surveying instruments. Turns are convenient when the important idea is a fraction or multiple of a complete revolution, such as 0.125 turn on a knob, 2.5 turns on a screw, or a full 360-degree animation.
These are circular angle units. They should not be confused with geographic coordinate formats. A longitude of -73.985428 degrees is an angular distance east or west of Greenwich, but it is also a coordinate component with a sign convention. Converting it to radians does not create a projected map coordinate. For latitude and longitude formatting, use the coordinates converter. For map-grid coordinates in meters, use the lat long to UTM converter.
Common pitfalls
- Treating radians as if they were degrees when using a programming language’s sine, cosine, or tangent functions.
- Reducing every angle to a 0 to 360 degree range when the number of rotations matters.
- Using gradians in a degree-based drawing without noticing that a right angle is 100 grad, not 90 grad.
- Assuming this tool handles arcminutes and arcseconds; those are sexagesimal subdivisions of a degree and are covered by DMS tools.
- Mixing math angles with coordinate signs. A negative math angle usually means clockwise or opposite orientation, while a negative longitude means west.
Sources
- NIST, Guide for the Use of the International System of Units — SI usage guidance for angle quantities and accepted units.
- NIST, SI units — overview of SI units, including the radian as the SI unit for plane angle.
- NOAA National Geodetic Survey, Datums — background on geodetic context that separates coordinate work from plain angle conversion.