What this calculator does
Every angle unit is a fixed fraction of a full turn, which makes the conversions exact rather than measured. A turn is 360 degrees, 2π radians or 400 gradians. A degree splits into 60 arcminutes and each of those into 60 arcseconds, the same sexagesimal scheme used for time.
The radian is the odd one out and the important one. It is defined so that an arc equal in length to the radius subtends one radian, which makes it about 57.3 degrees, a deliberately awkward number. The payoff is that calculus works cleanly in radians and not in degrees, which is why every trigonometric function in every programming language expects them.
The formula
The radian is the base unit here, and each other unit has an exact factor in radians derived from π: a degree is π/180, a gradian π/200, a full turn 2π, an arcminute π/10800 and an arcsecond π/648000. Converting multiplies by the factor of the source unit and divides by the factor of the target, so the relationships that should be exact come out exact.
| Term | Meaning |
|---|---|
| Degree | A 360th of a turn. The 360 is Babylonian and has no mathematical basis beyond dividing conveniently. |
| Radian | The angle subtending an arc equal to the radius, about 57.2958 degrees. The natural unit for calculus. |
| Gradian | A 400th of a turn, so a right angle is exactly 100. Designed to decimalise the circle and used in surveying. |
| Arcsecond | A 3,600th of a degree, used in astronomy and precision surveying where degrees are far too coarse. |
The inputs explained
| Field | What to enter |
|---|---|
| Value | The angle value to convert. |
| From | The unit it is currently in. |
| To | The unit you want the answer in. |
When to use it
Feeding an angle into code
Programming language trigonometric functions take radians almost without exception, while the angle you have is almost always in degrees. Getting this conversion wrong is one of the most common sources of nonsensical geometry output.
Reading a surveying or navigation figure
Surveying uses gradians in some countries and degrees in others, and navigation mixes degrees with arcminutes, where one arcminute of latitude is a nautical mile by definition.
Working with astronomical measurements
Angular sizes in astronomy are given in arcminutes and arcseconds because the objects are so small on the sky. The Moon is about 31 arcminutes across, which is barely half a degree.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What is one of each angle unit in degrees?
A single unit of each kind, converted to degrees to compare their size.
| From | In degrees |
|---|---|
| Radian | 57.29577951 deg |
| Gradian | 0.9 deg |
| Full turn | 360 deg |
| Arcminute | 0.01666667 deg |
| Arcsecond | 0.00027778 deg |
Converting 90 degrees to the other angle units
A right angle of 90 degrees expressed in each of the other units.
| To | Result |
|---|---|
| Radians | 1.57079633 rad |
| Gradians | 100 grad |
| Turns | 0.25 turn |
| Arcminutes | 5,400 arcmin |
Questions
How many degrees in a radian?
About 57.29578, which is 180 divided by π. The reverse, one degree, is π/180 radians or about 0.0174533. Neither is a round number and neither can be, since π is irrational.
Why do programming languages use radians?
Because the calculus is only clean in radians. The derivative of sine is cosine only when the angle is in radians; in degrees a factor of π/180 appears and propagates through every subsequent step. Since the underlying implementations are built on series expansions that assume radians, that is what the functions take.
What is a gradian for?
Decimalising the circle. A right angle is 100 gradians and a full turn 400, which makes mental arithmetic on quadrants easier. It came out of the same French revolutionary push that produced the metric system, and unlike the metre it never caught on widely, surviving mainly in surveying in a few countries.
Why are there 360 degrees in a circle?
It comes from Babylonian base-60 counting, and 360 has the practical virtue of dividing evenly by 2, 3, 4, 5, 6, 8, 9, 10, 12 and many more. There is nothing mathematically necessary about it, which is exactly why the radian exists.
For the same sexagesimal scheme applied to hours, minutes and seconds, see the time converter. For length and distance, see the length converter.