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ToolBoxGenie

Angle Converter

Converters · Added 15 August 2026

Convert angles between degrees, radians, gradians, full turns and the fine subdivisions used in astronomy and surveying. Factors here are exact multiples of π rather than rounded decimals, so a full turn converts to precisely 360 degrees and back again without drift.

Result

1 ° in radian

0.0174532925 rad

1 ° = 0.0174532925 rad

The same value in every unit

Degree (°)
1
Radian (rad)
0.0174532925
Gradian (gon) (gon)
1.11111111
Turn (revolution) (rev)
0.00277777778
Milliradian (mrad)
17.4532925
Arcminute (′)
60
Arcsecond (″)
3,600
Mil (NATO, 6400 per turn) (mil)
17.7777778

Radians are the natural unit for mathematics — a full turn is 2π, and the calculus of sine and cosine only comes out cleanly in them. Degrees survive because 360 divides evenly by so many numbers, which is convenient for navigation and construction.

How to use the angle converter

  1. 1Enter the angle you want to convert.
  2. 2Choose the unit it is in and the unit you want.
  3. 3The reference table shows the same angle in every unit at once.
  4. 4For trigonometry in code, remember most languages expect radians.
  5. 5Use the swap button to reverse the direction.

Examples

A right angle

Input
90°
Result
1.5708 rad · 100 gon · 0.25 turns

Gradians were designed so a right angle is exactly 100 — the metric system's attempt at angles.

Astronomical precision

Input
1 arcsecond
Result
0.000278° · 4.848 µrad

An arcsecond is 1/3600 of a degree, roughly the width of a coin seen from four kilometres away.

Programming a rotation

Input
π radians
Result
180° · 0.5 turns

Enter 3.14159265 to see this. Most programming languages take radians, not degrees.

About the angle converter

Three systems, three different design goals

Degrees were built for divisibility. With 360 of them in a circle, most fractions people actually need come out as whole numbers, which matters when you are working with a protractor, a sextant or a mental estimate rather than a calculator. Navigation and construction still run on degrees for exactly this reason.

Radians were built for mathematics. Defining the angle by the arc it cuts, in units of the radius, makes angle a pure ratio of two lengths — which is why radians are technically dimensionless. Every formula relating angles to arc lengths, angular velocities or oscillations simplifies in radians and acquires clutter in degrees.

Gradians were built for decimalisation, alongside the metre and the kilogram, and are the one part of that reform that largely failed. Their survival in European surveying is a reminder that unit choice is often settled by installed practice rather than by merit.

Where angle conversion goes wrong

The most frequent error in software is passing degrees to a function expecting radians. Nearly every programming language's sin, cos and tan take radians, and passing 90 instead of 1.5708 produces a plausible-looking number rather than an error — sin(90) in radians is about 0.894, which is not obviously wrong until the results are plotted. Spreadsheets are the usual exception, and even there the RADIANS function exists because the confusion is so common.

The second is dropping precision in the fine units. Arcminutes and arcseconds subdivide a degree by 60 and 3600, and coordinates written in degrees-minutes-seconds convert to decimal degrees with long fractional parts. Rounding that decimal too early moves a map position noticeably: a thousandth of a degree of latitude is about 111 metres.

The third is assuming military mils are milliradians. They are close but not equal — 6400 per turn against 6283.19 — and the difference is roughly 1.8%. At a thousand metres that is eighteen metres of error, which is more than enough to matter in the applications where mils are used at all.

Frequently asked questions

Why does mathematics use radians rather than degrees?
Because radians make the calculus of trigonometric functions come out cleanly. The derivative of sin(x) is cos(x) only when x is in radians; in degrees an awkward factor of π/180 appears in every derivative and integral. A radian is defined as the angle subtending an arc equal to the radius, which ties angle directly to arc length — arc length equals radius times angle, with no conversion factor. Degrees are a human convention; radians fall out of the mathematics itself.
Where did 360 degrees come from?
Almost certainly from Babylonian astronomy and its base-60 number system, combined with a year of roughly 360 days. The number survived because it is unusually convenient: 360 divides evenly by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120 and 180. That makes common fractions of a circle whole numbers, which matters enormously for navigation, construction and anything done without a calculator.
What is a gradian for?
It was the metric system's attempt to decimalise angles: 100 gradians in a right angle, 400 in a full turn. It never caught on generally but survives in surveying in parts of Europe, where it makes slope calculations tidy — a 1% slope is close to 1 gradian. Most scientific calculators still offer a GRAD mode, which is mainly encountered by accident when someone knocks it out of DEG and cannot work out why the answers are wrong.
What is a NATO mil?
An angular unit for artillery and rifle optics, defined as 1/6400 of a full turn. The appeal is that one mil subtends approximately one metre at a thousand metres, which turns range estimation into simple arithmetic. It is a rounded approximation of a milliradian — a true milliradian would give 6283.19 per turn — and other militaries have used 6000 or 6300 divisions for the same reason. Check which convention a piece of equipment uses.

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