Angle Converter
0.017453293 rad
- 1
- °
- 0.0174533
- rad
- 1.11111
- gon
- 0.00277778
- turn
- 60
- ′
- 3,600
- ″
- 17.7778
- mil
The angle converter handles seven units in both directions: degrees, radians, gradians, turns, arcminutes, arcseconds and NATO mils. Degrees to radians is the conversion most people arrive for, and it is the one where the constant is worth understanding rather than memorising.
How it works
radians = degrees x pi / 180
- pi / 180
- approximately 0.0174533, one degree expressed in radians
- 180 / pi
- approximately 57.29578, one radian expressed in degrees, for the reverse
- A radian is the angle where the arc length equals the radius. A full turn is 2π radians, which is why the constant is π/180 rather than something round.
- A gradian divides a right angle into 100 rather than 90, so a full turn is 400. It survives in some European surveying.
- An arcminute is 1/60 of a degree and an arcsecond 1/60 of that, used in astronomy and navigation.
- A NATO mil divides a turn into 6,400, chosen because at 1,000 metres one mil subtends almost exactly one metre.
Examples
Degrees to radians
Value
180
From
Degree
To
Radian
Result
3.1415927 rad
Half a turn is exactly π radians. Seeing π appear is the quickest check that a degrees-to-radians conversion is the right way round.
One radian in degrees
Value
1
From
Radian
To
Degree
Result
57.29578°
The reciprocal constant. If your result is 0.01745 when you expected 57, the conversion is inverted.
Frequently asked questions
Which way round is the degrees-to-radians formula?
Multiply by π/180 to get radians, multiply by 180/π to get degrees. The sanity check is magnitude: radians are smaller numbers, so 90 degrees becomes 1.57 radians, not 5,156.
Why do trigonometry functions in code expect radians?
Because the underlying mathematics is defined in radians. The derivative of sin(x) is cos(x) only when x is in radians. Math.sin in JavaScript, and its equivalent in nearly every language, takes radians, which is the most common cause of wildly wrong trigonometry results.
What is a gradian for?
It divides a right angle into 100 parts instead of 90, so percentages of a right angle are trivial. It saw real use in French surveying and still appears on scientific calculators as GRAD, which is why calculators sometimes give unexpected answers when left in that mode.
Why does a NATO mil divide the circle into 6,400?
Because it approximates a milliradian while dividing evenly. At 1,000 metres one mil subtends almost exactly one metre, which makes range and adjustment arithmetic simple in the field. A true milliradian would give 6,283.2 per turn, which does not divide neatly.