Degrees to Radians
Convert degrees to radians and back, with exact multiples of pi for the common angles — the form trigonometry and programming languages actually expect.
180° = π radians. A radian is the angle where the arc length equals the radius — about 57.3°.
| Degrees | Radians |
|---|---|
| 15 | 0.26 |
| 30 | 0.52 |
| 45 | 0.79 |
| 60 | 1.05 |
| 90 | 1.57 |
| 120 | 2.09 |
| 180 | 3.14 |
| 270 | 4.71 |
| 360 | 6.28 |
Every other conversion is a definition. This one is not.
An inch is exactly 25.4 millimetres. A foot is exactly 0.3048 metres. Those are not measurements that might be refined later; they are decisions, written down, with the digits ending where the definition says they end:
| Conversion | Exact? |
|---|---|
| 1 inch = 25.4 mm | yes, by definition |
| 1 foot = 0.3048 m | yes, by definition |
| 1 pound = 0.45359237 kg | yes, by definition |
| 1 US gallon = 3.785411784 L | yes, by definition |
| 1 degree = pi/180 rad | no, and never can be |
Degrees to radians is different in kind. The factor is π/180, and π is irrational — it has no exact decimal expansion, no exact binary expansion, and no representation in any floating point format that will ever exist. The number this page multiplies by is the closest available, and closest is the best anyone can do.
The effect is small and real. Converting every whole degree from 0 to 720 into radians and back, 672 of 721 return to exactly the number they started at. 49 do not — 15, 29, 30, 58, 59 and so on — landing a vanishing fraction away. Nothing you would see, since the display rounds, but it is the visible edge of a genuine difference.
A control worth having alongside it: the landmark angles all survive intact, so the failures are ordinary binary rounding rather than a bad factor.
| Degrees | Radians | Matches the exact expression |
|---|---|---|
| 30 | pi/6 | yes |
| 60 | pi/3 | yes |
| 90 | pi/2 | yes |
| 180 | pi | yes |
| 360 | 2 pi | yes |
A radian is a ratio, not a size
A radian is the angle where the arc length equals the radius: a length divided by a length. The
units cancel, so a radian is a pure number wearing a name. That is why you can write
sin(2) and mean something, and why a radian is not a unit the way a metre is.
It also decides which measure serious mathematics uses, and that can be checked numerically rather than taken on trust. Differentiating sine from first principles:
| Measured in | Numerical derivative | Cosine at that point | Ratio |
|---|---|---|---|
| radians, at 0.7 | 0.764842187 | 0.764842187 | 1 |
| degrees, at 40 | 0.013369998 | 0.766044443 | 0.017453292 |
In radians the derivative of sine is cosine, full stop. In degrees it is not — it picks up a stray constant, and that constant is 0.017453293, which is π/180 again. The factor turns up uninvited in every derivative you take and again in every integral, which is a tax nobody wants to pay.
And why a small angle is approximately itself
The same property gives the approximation that props up most of physics and engineering: for a small angle in radians, sin(x) is approximately x. Not roughly proportional to x — approximately equal to it, the same number.
| Degrees | Radians | Sine | Relative error |
|---|---|---|---|
| 1 | 0.017453 | 0.017452 | 0.0051% |
| 5 | 0.087266 | 0.087156 | 0.1270% |
| 10 | 0.174533 | 0.173648 | 0.5095% |
| 15 | 0.261799 | 0.258819 | 1.1515% |
| 20 | 0.349066 | 0.342020 | 2.0600% |
| 30 | 0.523599 | 0.500000 | 4.7198% |
10 degrees is the usual working limit, and the table shows why: it is where the error crosses half a per cent. By 30 degrees it is nearly five per cent, which is generally too much to wave away.
None of this works in degrees. The sine of ten degrees is 0.1736 and ten is ten — there is no sense in which they are close. The approximation is a fact about radians, which is another way of saying it is a fact about arc length.
How to use
- Type an angle in degrees or radians.
- Read the exact multiple of pi where one exists.
- Use the table for the standard angles.
- Check which unit your programming language's trig functions expect.
Frequently asked questions
What is the conversion?
Multiply degrees by pi and divide by 180 to get radians. Reverse it by multiplying radians by 180 and dividing by pi. One radian is about 57.2958 degrees.
What is a radian, really?
The angle you get when the arc length equals the radius. Wrap a piece of string the length of the radius around the circle's edge, and the angle it subtends at the centre is one radian. Since the circumference is 2 pi r, a full turn is 2 pi radians.
Why do mathematicians prefer radians?
Because calculus only works cleanly in them. The derivative of sine is cosine only when the angle is in radians; in degrees an awkward factor of pi over 180 appears everywhere. Radians are the natural unit, and degrees are the historical convention.
What are the common angles in radians?
30 degrees is pi/6, 45 is pi/4, 60 is pi/3, 90 is pi/2, 180 is pi, and 360 is 2 pi. These six cover most of what turns up in trigonometry exercises and are worth committing to memory.
Does my programming language use radians?
Almost certainly. The trigonometric functions in JavaScript, Python, C, Java and most others take radians. Passing degrees produces a result that is silently wrong rather than an error, which makes it one of the more annoying bugs to track down.
What are gradians?
A third system, where a right angle is 100 gradians and a full turn is 400. They appear on some calculators and in a little surveying work, and are otherwise a decimalisation attempt that never caught on. If your calculator gives a strange trig answer, check it is not in gradian mode.
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