Angle Converter: Degrees, Radians & Gradians
Convert between degrees, radians, gradians, arcminutes, arcseconds, and turns, instantly and precisely, right in your browser.
How the angle converter turns one number into six units, and why gradians exist at all
Six ways to measure the same turn of a wheel or the same slice of a circle sound redundant until you realize each one was invented for a different job. Degrees came from Babylonian astronomers carving a circle into 360 pieces. Radians came from calculus, where an angle equal to arc length over radius makes derivatives of sine and cosine behave. Gradians came from French revolutionary engineers who wanted a right angle to be a tidy 100 units instead of an awkward 90. This tool treats all six as equally valid and converts between any pair instantly.
The math runs entirely in your browser with plain JavaScript arithmetic. There is no server call, no API request, and nothing about the value you type ever leaves the page.
The conversion path: everything passes through radians
Rather than storing a separate formula for every possible pair of units, the tool picks one unit as the hub and routes every conversion through it. That hub is the radian, because it is the SI coherent unit for plane angle and every other unit in the table has a clean multiplicative relationship to it.
isNaN or an empty field stops the calculation and shows an inline error instead of a broken result.
Math.PI / 180, so typing 90 degrees produces 90 * (Math.PI/180), which is π/2 radians.
π/2 by the gradian factor Math.PI/200 gives 100 gradians, confirming a right angle is exactly 100 gon.
Because Math.PI is an irrational number truncated to double precision floating point, every conversion that touches degrees, gradians, arcminutes, arcseconds or turns carries a tiny rounding error in the last decimal place. Radians to radians is the only lossless conversion in the table, since the factor is exactly 1.
Full unit table, factor relative to one radian
| Unit | Symbol | Factor (× radian) |
|---|---|---|
| Degree | ° | π / 180 ≈ 0.0174533 |
| Radian | rad | 1 (base unit) |
| Gradian | gon | π / 200 ≈ 0.0157080 |
| Arcminute | ′ | π / 10800 ≈ 0.00029089 |
| Arcsecond | ″ | π / 648000 ≈ 0.00000485 |
| Turn (revolution) | rev | 2π ≈ 6.2831853 |
2 * Math.PI, is defined explicitly rather than derived from degrees, so a turn to degree conversion is really two multiplications chained through the radian hub, not a shortcut lookup.Where each unit actually gets used
Gradians in surveying
Land surveyors in continental Europe, particularly France and parts of the former Soviet bloc, still use gradians because a right angle at exactly 100 gon makes decimal calculations on total station equipment simpler than working with 90 degrees.
Arcseconds in astronomy and optics
Star catalogs record positions to fractions of an arcsecond, since one arcsecond is the apparent size of a coin viewed from several kilometers away. Telescope resolution and lens tolerances are routinely specified this way.
Turns in motors and robotics
Stepper motor and encoder documentation often expresses rotation in turns or fractions of a turn, since a full mechanical revolution is the natural unit for gear ratios and motion control firmware.
Naming conventions worth knowing
Degrees trace back roughly 4,000 years to Babylonian base-60 mathematics, which is also why a degree splits into 60 arcminutes and each arcminute splits into 60 arcseconds, mirroring hours, minutes and seconds of time. Gradians came out of the metric reforms of the 1790s, an attempt to decimalize angle the way meters decimalized length. The radian has no independent origin story in the same sense: it falls directly out of the geometric definition of arc length divided by radius, which is why the International System of Units treats it as dimensionless.
Doing this yourself
- BIPM SI Brochure is the official reference defining the radian as the coherent SI unit for plane angle.
- NIST Units and Systems of Measurement lays out the same conventions used domestically in the United States.
- International Astronomical Union, measuring the universe covers how arcseconds and arcminutes are used in celestial coordinate systems.
- MDN Math.PI reference documents the exact double precision value JavaScript uses for every trig based conversion here.
Where each angle unit is standard
CNC and 3D printer firmware configuration where a rotation needs converting between degrees and radians, CAD drawings imported from European survey data expressed in gradians, telescope mount alignment specified in arcseconds, robotics joint limits defined in turns, and classroom trigonometry homework where switching between degree mode and radian mode on a calculator produces a wrong answer until the units are converted properly first.
FAQ: Angle Converter
Multiply the degree value by π and divide by 180 (degrees × π / 180). For example, 90 degrees equals π/2 radians, approximately 1.5708. This formula comes from the fact that a full circle is both 360 degrees and 2π radians.
Nearly all math libraries (JavaScript’s Math.sin, Python’s math.sin, and so on) expect angles in radians because radians are the natural unit in calculus, a radian directly relates an angle to arc length on a unit circle, which simplifies derivatives and integrals of trigonometric functions. If you pass degrees into these functions without converting first, you’ll get mathematically incorrect results.
A gradian (or gon) divides a full circle into 400 equal parts instead of 360, making a right angle exactly 100 gradians, a deliberately round number under the metric system. Gradians are mainly used in surveying, some European civil engineering contexts, and certain calculators that include a dedicated ‘GRAD’ mode.
Arcminutes and arcseconds subdivide a degree the way minutes and seconds subdivide an hour: 1 degree equals 60 arcminutes, and 1 arcminute equals 60 arcseconds. They’re essential in astronomy and navigation for expressing very small angular measurements, like the apparent size of a distant star or a precise latitude/longitude coordinate.
One turn equals exactly 360 degrees, one full revolution around a circle. Turns are a convenient unit when working with rotational mechanics or animations, since expressing ‘1.5 turns’ is often more intuitive than ‘540 degrees’ for describing a rotation and a half.
Yes, all conversions use exact mathematical relationships based on π, computed with full floating-point precision in JavaScript, rather than rounded approximations. For extremely precision-critical applications, always verify against your field’s specific defined constants and rounding conventions.
From the blog
Reading around units
Where measurement systems came from and why two of them still argue.
