Angle Converter
Convert between degrees, radians, gradians, arcminutes, arcseconds, and revolutions.
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How to use this calculator
- 1
Enter the angle value you want to convert.
- 2
Select the unit you are converting from (e.g. degrees).
- 3
Select the target unit (e.g. radians).
- 4
The result appears instantly.
Frequently asked questions
Which angle units does this converter support?
Degrees, radians, gradians, milliradians, revolutions, arcminutes and arcseconds. Each is stored relative to the degree, so a conversion between any two units is exact apart from the rounding of the displayed result to about ten significant digits.
How do I convert degrees to radians by hand?
Multiply by π ÷ 180, which is about 0.0174533. So 45° is 0.7853982 rad and 180° is 3.1415927 rad. In the other direction multiply radians by 180 ÷ π, about 57.29578, which is why one radian is roughly 57.3°.
What exactly is a radian?
One radian is the angle at the centre of a circle that cuts off an arc whose length equals the radius. Because a circumference is 2π times the radius, a full turn is 2π rad, or 6.283185 rad. Calculus formulas such as the derivative of sin x only take their simple form when x is in radians.
Why does my calculator give a wrong sine or cosine?
Almost always because it is in the wrong angle mode. Entering 90 while in radian mode treats it as 90 rad, not 90°. Convert first (90° is 1.570796 rad) or switch the mode, and check which one your spreadsheet or programming language expects; most programming languages use radians.
What is a gradian, and where is it used?
A gradian, also called a gon, is 1/400 of a full circle, so a right angle is exactly 100 grad and 270° is 300 grad. Its appeal is that quarter turns are round numbers, and it survives mainly in surveying instruments. This tool uses 1 grad = 0.9°.
Is the milliradian here the same as a military mil?
No. A milliradian is one thousandth of a radian, so a circle holds about 6,283.185 of them. Several military systems use rounded mils instead, for example NATO’s 6,400 per circle, so scope or artillery data in mils may not match this value exactly. Check which definition your equipment uses.
How are arcminutes and arcseconds related to degrees?
One degree is 60 arcminutes and one arcminute is 60 arcseconds, so a degree holds 3,600 arcseconds. They are sexagesimal subdivisions, not decimals: 12.5 arcminutes is 0.208333°, whereas 12°30′ would mean 12.5°. The tool takes decimal input, so convert a degrees-minutes-seconds reading in steps.
Does it handle angles beyond one revolution or negative angles?
Yes, the conversion is linear, so 720° gives 2 revolutions and −90° gives −100 grad. It does not wrap results into a 0 to 360° range or work out compass bearings, and it does not do trigonometry.
Angle units: degrees, radians, gradians and the rest
One circle, several ways to cut it
A full turn is 360° in everyday geometry, 2π radians in mathematics, 400 gradians in some surveying work and 1 revolution when you are counting rotations. All seven units in this converter measure the same thing and differ only in where the scale starts counting. The table lists the angles people look up most often.
| Degrees | Radians | Gradians | Revolutions |
|---|---|---|---|
| 30° | 0.52359878 | 33.33333333 | 0.083333333 |
| 45° | 0.78539816 | 50 | 0.125 |
| 60° | 1.047197551 | 66.66666667 | 0.16666667 |
| 90° | 1.570796327 | 100 | 0.25 |
| 180° | 3.141592654 | 200 | 0.5 |
| 270° | 4.71238898 | 300 | 0.75 |
| 360° | 6.283185307 | 400 | 1 |
Worked example: a radian answer from a physics problem
Suppose a rotation comes out as 2.5 rad and you want a protractor reading. The converter returns 143.2394488°, so about 143.2°. Going the other way, 1 mrad equals 206.2648062 arcseconds, which is why a 1 mrad error in aiming shifts the point of impact by roughly 1 m at a distance of 1 km.
Worked example: small angles in astronomy and optics
Fine angles are often quoted in arcminutes. A 12.5 arcminute field of view converts to 0.20833333°, and one arcsecond is 0.00027777778°, so a degree of sky covers 3,600 of them. For the reverse, 0.5° is exactly 30 arcminutes.
Choosing a unit
Use radians for calculus, physics and most programming libraries, because arc length equals radius times angle only in radians. Degrees suit drawings, construction and navigation by bearing. Gradians appeal when a right angle should be a round 100. Arcminutes and arcseconds suit astronomy and fine optical work, where a degree is too coarse. Revolutions are natural for motors and dials.
Using the converter step by step
Type the angle into the first box, pick its unit, and choose the target unit; either box can be edited and the other follows. The page also lists the value in every other unit, so one entry gives the radian, gradian and revolution forms together. Negative angles and values above one revolution work because the factors are linear. If your answer must lie between 0° and 360°, subtract whole revolutions yourself. Results are shown to about ten significant digits, enough for any engineering use, so round to the precision your measurement actually had. Rounding is the one place errors creep in.
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Results are estimates for informational purposes only and do not constitute professional financial, medical, legal, or technical advice. Read full disclaimer →