Angle converter
Type an angle and pick its unit: degrees (also with minutes and seconds), radians, gradians and turns update as you type, with the exact value in terms of pi whenever one exists.
| Degrees | Radians | Gradians | Turns |
|---|---|---|---|
| 0° | 0 | 0 gon | 0 |
| 30° | π/6 | 33.33 gon | 1/12 |
| 45° | π/4 | 50 gon | 1/8 |
| 60° | π/3 | 66.67 gon | 1/6 |
| 90° | π/2 | 100 gon | 1/4 |
| 120° | 2π/3 | 133.33 gon | 1/3 |
| 135° | 3π/4 | 150 gon | 3/8 |
| 180° | π | 200 gon | 1/2 |
| 270° | 3π/2 | 300 gon | 3/4 |
| 360° | 2π | 400 gon | 1 |
Degrees, radians, gradians and turns: when each is used
Degrees (a full turn = 360°) are the everyday unit, the one you see in school geometry, on protractors, and in weather reports about wind direction. Radians (a full turn = 2π) are the standard in math and programming: the trigonometric functions in almost every language, JavaScript included, only work in radians. Gradians, or "gon" (a full turn = 400), survive mainly in surveying and in the total stations used by land surveyors, because they turn a right angle into a convenient 100.
The turn, finally, is the simplest unit: a full turn is 1 and a right angle 0.25. CSS uses it (the «turn» unit) as do several graphics programs, and it is the most natural way to say «three quarters of a turn».
Degrees, minutes and seconds
A degree is split into 60 minutes and a minute into 60 seconds, like an hour: 45° 30′ 15″ means 45 degrees, half a degree and fifteen seconds, that is 45.504167°. It is how nautical charts, surveyors' instruments and astronomy write angles.
In the field the angle is written as it comes: 45° 30′ 15″, 45°30'15", 45 30 15 or 45:30:15, with a minus in front for a negative angle. Minutes and seconds go from 0 to 59, and only the last number may have decimals. The result in degrees, minutes and seconds goes to the hundredth of a second, and the total is rounded first: so an angle never comes out as 30′ 60″.
The conversion, and the number to remember
Half a turn is 180° = π radians: everything comes from there. A full turn is 360° = 2π; a right angle is 90° = π/2 ≈ 1.5708; 60° is π/3 ≈ 1.0472; 45° is π/4 ≈ 0.7854; 30° is π/6 ≈ 0.5236. For mental arithmetic: one radian is a little under 57.3 degrees, a fairly wide angle, more than half a right angle.
The formulas are radians = degrees × π ÷ 180, gradians = degrees × 10 ÷ 9 and turns = degrees ÷ 360. When the angle in degrees is a whole number, or has a simple decimal part such as 22.5 or 7.5, the converter also shows the exact value in terms of pi (45° = π/4, 22.5° = π/8): for working by hand it is far more useful than the decimal 0.785398.
Why radians exist
Because the degree is a convention (the 360 degrees come from the Babylonians, not from geometry) while the radian is a natural measure: an angle of one radian is the one that cuts out on the circle an arc as long as the radius. The convenience follows from that definition: the length of an arc is simply radius × angle, with no conversion factor, and in physics rotation speed is written the same way.
The mistake everyone makes the first time
In programming languages the sine, cosine and tangent functions want radians, not degrees, and they don't say so: passing 90 instead of 1.5708 doesn't raise an error, it produces a wrong number that looks plausible. It applies to JavaScript, Python, C, Java and to spreadsheets. If a drawing turns a little when it should turn a quarter, that's nearly always the cause.
There's a third unit that shows up on scientific calculators too, the gradian (400 to a turn, the GRAD key): if your trigonometry results are all slightly odd, check the calculator isn't on that instead of DEG.
Nearby tools
For geographic coordinates, with north, south, east and west, there is GPS coordinates. For sums with sines and cosines the page is Scientific calculator, and for triangles Solve a triangle.