Free Angle Converter Online

An angle converter takes a rotational measurement in one system — degrees from a protractor, radians from a calculus formula, gradians from a surveying instrument, arcminutes or arcseconds from an astronomy chart, full turns from a mechanical spec — and restates it in whichever of the others you need next. This one covers six units: degree, radian, gradian, arcminute, arcsecond, and turn, and because degrees, gradians, and turns are all just different ways of slicing up the same full circle, every conversion here is an exact multiplication with no rounding built into the underlying factor.

How the Angle Converter Works

Enter a number into the single input field — the tool reads it as degrees, the unit most people think in by default — and the output pane fills in within roughly 300 milliseconds of you pausing, showing that same angle restated in radians, arcminutes, arcseconds, turns, and gradians in one pass. Auto-convert runs continuously out of the box, though a switch in the header lets you turn it off and trigger the recalculation manually with a button instead, handy if you're pasting in a longer decimal figure. A Copy button grabs the whole output for pasting elsewhere, a Save button writes it to a plain-text file, and a Clear control wipes both fields clean. Nothing here needs an account. The same six-unit math is also available as a metered API call for anyone scripting geometry or CAD-adjacent calculations in bulk — full parameters are documented at /docs.

As a worked example: typing 45 into the field — 45 degrees — returns approximately 0.7854 radians, 2,700 arcminutes, 162,000 arcseconds, 0.125 turns, and exactly 50 gradians, every figure computed the instant the number lands in the field from the exact factors in the table further down.

Why People Convert Angle Units

An angle rarely stays in one unit for long once it crosses from one field or piece of software into another:

• Trigonometry and calculus coursework. Most formulas are written in radians, but problems are frequently stated in degrees, so translating between the two is a routine first step before any calculation can even start. A right angle of 90 degrees, for instance, converts to exactly π/2 radians, about 1.5708, a figure that shows up constantly in trigonometric identities.
• Surveying and civil engineering. Some instruments and older European land-survey records use gradians (also called gons) instead of degrees, and reconciling a modern survey with historical records means converting between the two systems. A gradian-based bearing of exactly 133.333 gradians, for instance, converts to precisely 120 degrees once cross-referenced against a modern degree-based GPS reading.
• Astronomy and navigation. Sky coordinate systems are commonly expressed down to arcminutes and arcseconds, a level of precision far finer than whole degrees allow when locating a faint or distant object. A star's angular separation of 3,600 arcseconds, for example, is exactly 1 degree — fine enough to distinguish features a whole-degree measurement would blur together entirely, which is why star charts and telescope mounts are calibrated down to arcseconds rather than stopping at whole degrees.
• Programming graphics, robotics, or game physics. Rotation functions in most languages and engines expect radians as input even when the original design spec was written entirely in degrees. A 30-degree rotation called for in a design spec, for instance, needs to become approximately 0.5236 radians (π/6) before most graphics or physics libraries will accept it as a valid argument.
• Mechanical and CNC work. A full rotation is sometimes specified as "turns" rather than a degree value, particularly for lead screws and stepper-motor movements, and matching that against a degree-based drawing takes a quick conversion. A lead-screw spec calling for 2.5 turns, for instance, corresponds to exactly 900 degrees of rotation — two and a half full revolutions translated directly into the value a CAD drawing would expect, and a figure that's easy to get wrong by a whole revolution if the conversion is done from memory instead of checked.

Understanding How Each Angle Unit Is Defined

Unit	Equivalent in degrees	Notes
Degree	1	1/360 of a full circle — the everyday default
Radian	57.2957795131	180/π degrees; the angle where an arc length equals the radius, which is why calculus formulas for sine and cosine come out clean in this unit
Gradian	0.9	1/400 of a full circle, also called a "gon," used in surveying and some European civil-engineering contexts
Arcminute	0.0166666667	1/60 of a degree
Arcsecond	0.000277777778	1/3600 of a degree, or 1/60 of an arcminute
Turn	360	One full revolution, sometimes called a "revolution" or "cycle"

The 360-degree circle itself traces back to ancient Babylonian mathematics, which used a base-60 counting system alongside a calendar of roughly 360 days; 360 divides evenly by a large number of whole numbers (2, 3, 4, 5, 6, 8, 9, 10, 12, and more), which made it a practical choice for hand calculations long before calculators existed. That inherited convention is why degrees, rather than something rounder like 100 or 1,000 units per circle, are still the everyday default.

A handful of angles come up often enough in geometry and design work that it helps to have their equivalents on hand without typing each one in separately — every figure below is exactly what entering the degree value into this converter would return:

Degrees	Radians (approx.)	Turns	Gradians
30°	0.5236	0.0833	33.33
45°	0.7854	0.125	50
60°	1.0472	0.1667	66.67
90°	1.5708	0.25	100
180°	3.1416	0.5	200
270°	4.7124	0.75	300
360°	6.2832	1	400

Approach	Tradeoff
Working in degrees	Intuitive and matches how protractors, compasses, and everyday instructions are labeled, but most calculus and physics formulas require converting to radians first
Working in radians	The natural unit for calculus, since derivatives and integrals of sine and cosine only come out clean here, but the numbers are unintuitive to read at a glance — a right angle is about 1.5708, not a round figure
Working in gradians	Splits a right angle into a clean 100 units, convenient for certain surveying calculations, but it's a niche unit outside those fields and most software defaults elsewhere
Working in arcminutes or arcseconds	Necessary for astronomy-level angular precision, but far too fine-grained a unit for describing everyday rotations

Regardless of which unit ends up on the page or the drawing, the tool always computes through the same degree-based factors in the table above, so picking a comfortable display unit never costs anything in precision — the underlying radian, gradian, and arcsecond figures stay exact either way.

Angle Converter vs. Doing It Manually

• Reaching for mental math. Converting to radians means multiplying by π/180, an irrational number you can only ever approximate, so a manual conversion is a rounded guess unless you carry many decimal places — and carrying enough decimal places by hand defeats the point of doing it quickly.
• A calculator's DEG/RAD/GRAD mode. It works, but converts one value at a time, and switching modes on a physical or app-based calculator is easy to forget partway through a problem, which is exactly how a correctly-typed number turns into a wrong answer.
• A programming language's built-in function. Most languages ship a radians() or toRadians() helper, useful inside code but not something you'd reach for just to check a single number by hand without opening an editor or a console first.
• This tool. Every unit updates from one typed number, with no mode-switching step to remember or forget, and the same conversion is available through the API for anyone converting angles inside a script instead of by hand.

How Your Input Is Processed

Whatever angle you type is converted directly in your browser using JavaScript running on your device — the value never leaves your machine to reach a server, since the entire six-unit calculation happens locally the instant you stop typing. No sign-in is required, and the tool holds nothing in memory once you clear the field or leave the page.

Common Questions About Converting Angle Units

Why does calculus use radians instead of degrees?

Because the derivative of sin(x) works out to exactly cos(x) only when x is measured in radians; using degrees introduces an extra scaling constant into every derivative and integral, and that nuisance is enough of a reason that radians became the default unit throughout higher mathematics.

What exactly is a gradian, and who still uses it?

A gradian, or "gon," splits a full circle into 400 equal parts instead of 360, so a right angle comes out to a clean 100 gradians. It shows up mainly in surveying and geodesy, particularly in parts of continental Europe, and appears on some older theodolites and total-station instruments, which is exactly the situation where reconciling a gradian-based bearing against a degree-based GPS coordinate becomes a routine, if occasional, necessity.

How many arcseconds are in one degree?

3,600 — a degree splits into 60 arcminutes, and each arcminute splits into 60 arcseconds, so 60 × 60 gives 3,600 arcseconds per degree. That level of precision matters in astronomy, where the apparent size of a distant object is often measured in arcseconds rather than whole degrees.

Is a "turn" the same thing as 2π radians?

Yes — a turn is defined as exactly one full revolution, which is 360 degrees and also exactly 2π radians, roughly 6.28319. It's a convenient unit for counting whole rotations, like the number of times a wheel, gear, or motor shaft spins.

Why is a circle 360 degrees instead of a rounder number?

The convention comes from ancient Babylonian mathematics and its base-60 counting system, paired with a calendar of roughly 360 days. Because 360 divides evenly by so many small whole numbers, it made hand calculation far more practical for the astronomers and surveyors who established the convention long before calculators existed.

Can this converter handle angular velocity or rotational speed instead of a static angle?

No — angular velocity is an angle per unit of time (radians per second, for instance), which is a rate rather than a fixed angle, and this tool only converts static angle values. For a shaft or wheel's rotational speed in rpm or Hz, the Frequency Converter is the correct tool, since angular velocity is derived by multiplying rotational frequency by 2π rather than by converting an angle unit on its own.

Related Tools

Angles rarely show up in isolation — geometry problems that need an angle conversion usually need a linear measurement alongside it, which the Length Converter covers, while circular or sector-area calculations pair naturally with the Area Converter. Anything involving rotational speed, such as motor rpm or wheel spin rate, ties angle and time together, which is exactly what the Frequency Converter and the Time Unit Converter are built to handle. Between the four of them, most everyday and technical angle problems — from a single classroom trigonometry question to a full CAD or robotics rotation spec — end up covered without needing a fifth reference source.