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Angle Converter
Convert an angle between degrees, radians, gradians, full turns, arcminutes, arcseconds and milliradians. Pick the two units and read the same angle in every other unit below the result.
Where each angle unit comes from
A full circle can be divided in several ways, and each unit here is a fixed fraction of one turn.
- Degree: 1/360 of a turn, so a right angle is 90°.
- Radian: the angle at the centre of a circle that cuts off an arc equal in length to the radius. A full turn is 2π radians, so 1 rad = 180/π degrees ≈ 57.2958°.
- Gradian: 1/400 of a turn, so a right angle is 100 grad. It appears mainly in surveying.
- Arcminute and arcsecond: 1/60 and 1/3,600 of a degree, used in astronomy, navigation and mapping.
- Milliradian: one thousandth of a radian, used in optics and ranging.
Common angles
| Degrees | Radians | Gradians | Turns |
|---|---|---|---|
| 30° | π/6 | 33.33 | 1/12 |
| 45° | π/4 | 50 | 1/8 |
| 60° | π/3 | 66.67 | 1/6 |
| 90° | π/2 | 100 | 1/4 |
| 180° | π | 200 | 1/2 |
| 360° | 2π | 400 | 1 |
Decimal degrees and degrees-minutes-seconds
Maps and coordinates often use degrees, minutes and seconds. To convert a decimal angle, keep the whole degrees, multiply the fractional part by 60 to get minutes, and repeat for the seconds. For 40.7128°, the fraction 0.7128 × 60 = 42.768′, then 0.768 × 60 = 46.08″, giving 40° 42′ 46.08″. To go back, compute degrees + minutes/60 + seconds/3,600. Enter arcminutes or arcseconds into this converter to turn them into decimal degrees.
Mistakes to avoid
- Wrong calculator mode. Trigonometry functions expect radians or degrees depending on the mode. sin(90) is 1 in degree mode but about 0.894 in radian mode.
- Confusing slope with angle. A 100% grade is 45°, not 90°, and a 1° incline is about 1.75%.
- Forgetting direction. Compass bearings run clockwise from north, while mathematical angles usually run anticlockwise from the positive x axis.
Angles larger than one turn, or negative angles, are accepted and converted as entered, so 450° is 1.25 turns and −90° is −π/2 radians. The tool does not wrap values into the range 0 to 360. If you need the equivalent angle within one turn, subtract or add whole turns yourself.
Everyday uses include setting a protractor-based design in software that expects radians, reading survey bearings recorded in gradians, converting latitude and longitude between decimal and sexagesimal notation, and checking the small-angle sizes quoted for telescopes and rifle scopes, where one milliradian spans about one metre at a distance of one kilometre.
Worked example
A right angle is 90 degrees. What is it in radians?
Inputs
| Value | 90 |
|---|---|
| From | Degrees (°) |
| To | Radians (rad) |
Result
90 ° = 1.570796 rad
| Gradians (grad) | 100 |
|---|---|
| Turns (turn) | 0.25 |
| Arcminutes (′) | 5,400 |
| Arcseconds (″) | 324,000 |
| Milliradians (mrad) | 1,570.7963 |
- Factors are exact definitions where one exists; results are shown to about six significant digits.
Accuracy note: Factors use the standard definitions stated on this page. Results are rounded for display; for engineering or legal work use the full-precision factors.
Frequently asked questions
How many degrees are in a radian?
One radian is 180/π degrees, which is about 57.2958 degrees, so a full circle of 360 degrees is 2π radians.
How do I convert degrees to radians?
Multiply the degrees by π and divide by 180. For example, 90° × π ÷ 180 = π/2, about 1.5708 radians.
How many gradians are in a right angle?
A right angle is exactly 100 gradians, since a full circle is 400 gradians.
How many arcseconds are in a degree?
There are 3,600 arcseconds in a degree: 60 arcminutes, each made of 60 arcseconds.
Why does my calculator give a strange sine value?
The calculator is probably in the wrong angle mode. Convert the angle to the unit its mode expects, or switch the mode.
Related calculators
More conversion references
For angle conversion tables beyond this tool, see PakMath angle conversion tables. PakMath is a sister project of Adzita.