What a radian actually measures
A radian is the angle you sweep out when the arc you travel along a circle is exactly as long as that circle's radius. Walk one radius-length around the rim of any circle, look back at the centre, and the angle you have turned through is one radian. Because the definition compares two lengths, the radian carries no units of its own: it is a pure ratio, which is why it slots into formulas without dragging a conversion factor behind it.
That property is the whole reason radians exist. The circumference of a circle is 2πr, so a full revolution contains 2π radius-lengths of arc, which is 2π ≈ 6.283185 radians. A degree, by contrast, is an arbitrary 1/360 of a revolution inherited from Babylonian astronomy; 360 is convenient because it factors so many ways, but it has no geometric meaning. Every derivative, series expansion and small-angle approximation in calculus is written for radians, and every one of them acquires a stray factor of π/180 if you feed it degrees instead.
The practical consequence shows up the first time you call a trigonometric function in code. Python's math.sin, JavaScript's Math.sin, C's sin and R's sin all take radians. Spreadsheets do too: Excel and Google Sheets both require RADIANS() around a degree value before SIN() will give the right answer. Hand-held scientific calculators are the exception, and their DEG/RAD mode switch is the single most common source of a wrong trigonometry answer.
Why the conversion factor is π/180
You derive the factor by writing one full turn in both units and dividing. A full turn is 360°, and a full turn is 2π radians, so 360° = 2π rad. Divide both sides by 360 and you get 1° = 2π/360 rad = π/180 rad. Multiplying any degree measure by π/180 therefore rescales it into radians, and the numerical value of π/180 is 0.017453292519943295.
Reading the same identity the other way gives the inverse conversion, 1 rad = 180/π degrees ≈ 57.29577951°. That number is worth committing to memory, because it lets you sanity-check any conversion in your head: a radian is a bit less than 60°, so an answer in radians should always be a little larger than the degree value divided by 60. Our radians to degrees calculator performs the reverse trip.
Gradians follow the same pattern with a different base. A gradian, also written gon, splits the full turn into 400 parts so that a right angle is exactly 100 gon. That makes the degree-to-gradian factor 400/360 = 10/9. European surveying instruments and some civil-engineering software still work in gon, so the conversion appears whenever survey data crosses a border. Turns, the fourth unit reported here, are simply the angle divided by 360; they are the natural unit for rotational counts, gear ratios and phase in signal processing.
Worked example: converting a 137°30'36" bearing
A total station gives you a horizontal angle of 137°30'36". Convert it to radians for a coordinate-geometry routine.
- Convert the minutes. 30 arcminutes ÷ 60 = 0.5°.
- Convert the seconds. 36 arcseconds ÷ 3600 = 0.01°.
- Add up the decimal degrees. 137 + 0.5 + 0.01 = 137.51°.
- Multiply by π/180. 137.51 × 0.017453292519943295 = 2.40012814 rad.
- Check the magnitude. 137.51 ÷ 57.29578 = 2.4001, which agrees, so the arithmetic is sound.
The same angle in gradians is 137.51 × 10/9 = 152.789 gon, and as a fraction of a revolution it is 137.51 ÷ 360 = 0.381972 turns. The exact form still reduces, because 137.51° is a whole number of arcseconds: 137.51 × 3600 = 495,036 arcseconds, and a half turn is 648,000 arcseconds, so the angle is 495036π/648000 rad. Divide both numbers by their common factor of 36 and the exact value field reports 13751π/18000, whose decimal value 0.763944π agrees with the radian answer above.
Now take a textbook angle instead. For 135°, the ratio 135/180 reduces: the greatest common divisor of 135 and 180 is 45, so 135/180 = 3/4, and the exact answer is 3π/4 rad = 2.35619449 rad. That reduction is exactly what the exact value output performs for you.
How to read the result
Check the exact π form first. If your angle is one of the standard classroom angles — any multiple of 15° — the exact form is the answer a marker wants to see, and the decimal is only a cross-check. Any angle that is a whole number of arcseconds still gets an exact fraction, but it can be an unwieldy one: 137.51° reduces to 13751π/18000 rather than to anything you would want to write on a diagram. Only when the angle is not a whole number of arcseconds does the field fall back to a decimal coefficient such as 0.763944π, and at that point the decimal radian value is the useful one.
Then check the magnitude. Any angle inside one revolution converts to a radian value between 0 and 6.2832. If you see a radian value above about 6.3, your angle exceeded a full turn, and you should decide whether you want the unwound angle or its coterminal equivalent inside 0–360°; the coterminal angle calculator handles that reduction. Angles of thousands of degrees are legitimate in rotational mechanics, where the total swept angle matters, but are usually a data-entry slip in geometry.
Finally, watch the sign. A negative degree value produces a negative radian value, and every trigonometric function accepts it happily. In standard position a negative angle is measured clockwise from the positive x-axis, so −90° and +270° point the same way even though their radian values, −π/2 and 3π/2, differ by a full 2π.
Common angles in degrees, radians, gradians and turns
| Degrees | Exact radians | Radians (decimal) | Gradians | Turns |
|---|---|---|---|---|
| 0° | 0 | 0.000000 | 0 | 0 |
| 15° | π/12 | 0.261799 | 16.667 | 0.041667 |
| 30° | π/6 | 0.523599 | 33.333 | 0.083333 |
| 45° | π/4 | 0.785398 | 50 | 0.125 |
| 60° | π/3 | 1.047198 | 66.667 | 0.166667 |
| 90° | π/2 | 1.570796 | 100 | 0.25 |
| 120° | 2π/3 | 2.094395 | 133.333 | 0.333333 |
| 135° | 3π/4 | 2.356194 | 150 | 0.375 |
| 150° | 5π/6 | 2.617994 | 166.667 | 0.416667 |
| 180° | π | 3.141593 | 200 | 0.5 |
| 270° | 3π/2 | 4.712389 | 300 | 0.75 |
| 360° | 2π | 6.283185 | 400 | 1 |
One arcminute is 0.000290888 rad and one arcsecond is 0.0000048481 rad, both derived from the same π/180 factor.
Mistakes that produce a wrong angle
- Leaving a calculator in the wrong mode. A pocket calculator set to DEG returns 0.0087 for sin(0.5) when you wanted the sine of half a radian, 0.4794. Check the DEG/RAD indicator before every trigonometry problem.
- Treating decimal minutes as decimal degrees. A bearing of 42°30' is 42.5°, not 42.30°. The gap is 0.2°, which is 3.5 metres of offset at one kilometre.
- Rounding π/180 too early. Using 0.0175 instead of 0.0174532925 puts a 0.27% error into the result, enough to move a survey point by 27 centimetres in 100 metres.
- Assuming a programming language takes degrees. Every mainstream language's trigonometric library works in radians; spreadsheets need an explicit
RADIANS()wrapper. - Dropping the sign when converting DMS. A latitude of −33°51'54" means 33 degrees, 51 minutes and 54 seconds south; the minus applies to the whole reading, not just the degree part.
What the SI says about the radian
The BIPM's SI Brochure, 9th edition, lists the radian as the coherent SI unit of plane angle, defined as the ratio of arc length to radius, with the symbol rad and the dimensionless derived unit m/m. The degree, the arcminute and the arcsecond are not SI units, but the Brochure accepts them for use with the SI because of their entrenched role in navigation, astronomy and surveying. The gradian appears in the same table of accepted non-SI units. Nothing in the SI forbids degrees; it simply makes the radian the unit that formulas are written for.
Where the conversion matters most
Calculus is the first place the choice of unit stops being cosmetic. The derivative of sin x is cos x only when x is in radians; in degrees it becomes (π/180)cos x. The same factor infects the Taylor series, so the familiar small-angle approximation sin θ ≈ θ holds only for radians. At 5° = 0.0872665 rad, sin θ = 0.0871557, an error of 0.13%; write 5 instead and the approximation is nonsense.
Rotational mechanics is the second. Angular velocity in rad/s multiplies directly by radius to give tangential speed in m/s, and angular acceleration in rad/s² multiplies by radius to give tangential acceleration. Do the same with degrees per second and every answer is off by the factor 57.29578. Converting a motor's rating from revolutions per minute to rad/s uses the turn output here: 1 rpm = 2π/60 rad/s = 0.10471976 rad/s.
Geometry is the third, and the cheapest to get right. Arc length is s = rθ and sector area is A = ½r²θ, both with θ in radians; feed degrees to either and the answer is 57 times too large. Our arc length calculator and circular sector area calculator handle the unit switch internally, and the reference angle calculator is the tool to reach for when you need the acute angle a large rotation is equivalent to. Once your angle is in radians, feed it straight into the sin cos tan calculator.