An angle converter for maths, maps, surveying and sights
This angle converter moves between degrees, radians, gradians (gons), turns, arcminutes, arcseconds, NATO mils and milliradians. Type an angle once and see it in all eight units, with the conversion formula — for example rad = ° × 0.0174532925199.
Underneath the main converter is a second tool for map and GPS work: decimal degrees ⇄ degrees, minutes and seconds (DMS). Paste a coordinate such as 40.446111 to get 40° 26′ 46″, or type degrees, minutes and seconds to get the decimal value that mapping apps and spreadsheets expect.
How to use the Angle Converter
- Type an angle in the Value box. Fractions work, and the ± button makes it negative.
- Choose the unit you have in From — degrees, radians, gradians, turns, arcminutes, arcseconds, mils or milliradians.
- Choose the unit you want in To, or press the round swap button to reverse.
- Read the result and the formula, set Precision (radians often need 10 or more digits) and press Copy.
- For coordinates, use the Decimal degrees ⇄ degrees, minutes, seconds panel: type in either side and press Copy DMS for the formatted result.
Angle Converter features
Eight angle units
Degrees, radians, gradians, turns, arcminutes, arcseconds, NATO mils and true milliradians.
Two-way DMS tool
Convert 51.4778° to 51° 28′ 40.08″ and back, with negative values for south and west.
Full-precision radians
π-based conversions are shown to up to 15 significant digits for programming and engineering.
Shooting and artillery units
NATO mils (6,400 per turn) and milliradians (about 6,283 per turn) are separate units, not one vague “mil”.
Every unit at once
See an angle in all units together and tap any row to make it the target.
Angle units and their definitions
A full turn is the one thing every angle unit agrees on. Degrees split it into 360 parts, a choice inherited from Babylonian base-60 astronomy. Radians measure an angle by the arc it cuts on a circle compared with the radius, so a full turn is 2π radians; the radian is the SI unit and the natural one for calculus and physics. Gradians, introduced in France with the metric system, give 100 to a right angle and are still used in some surveying instruments.
| Unit | Per full turn | In degrees |
|---|---|---|
| Degree (°) | 360 | 1 |
| Radian (rad) | 2π ≈ 6.283185 | 180/π ≈ 57.29578 |
| Gradian / gon (grad) | 400 | 0.9 |
| Turn (revolution) | 1 | 360 |
| Arcminute (′) | 21,600 | 1/60 |
| Arcsecond (″) | 1,296,000 | 1/3,600 |
| NATO mil | 6,400 | 0.05625 |
| Milliradian (mrad) | 2,000π ≈ 6,283.185 | ≈ 0.0572958 |
Worked examples
Degrees to radians — multiply by π/180. 45° × π/180 = π/4 = 0.785398 rad, and 30° = 0.523599 rad.
Radians to degrees — multiply by 180/π. 1 rad = 57.29578°, and 2.5 rad = 143.239°.
Decimal degrees to DMS. Take 40.446111°. The whole number, 40, is the degrees. Multiply the remainder by 60: 0.446111 × 60 = 26.76667, so 26 minutes. Multiply the new remainder by 60: 0.76667 × 60 = 46 seconds. The answer is 40° 26′ 46″.
DMS to decimal degrees. 73° 58′ 48″ = 73 + 58/60 + 48/3,600 = 73.98°. If the coordinate is west or south, make it negative: −73.98.
Trigonometry functions on a scientific calculator expect either degree or radian mode; converting first with this tool avoids the classic wrong-mode error.
Angles in the real world
- GPS and maps: one arcminute of latitude is about one nautical mile (1,852 m), so one arcsecond is roughly 31 m. Five decimal places of a degree pin a point to about a metre.
- Shooting optics: one minute of angle (MOA) spans about 1.047 inches at 100 yards; one milliradian spans 10 cm at 100 m.
- Military mils: NATO uses 6,400 mils per circle; other armies have used 6,000 or 6,300, so check which system a map or compass uses.
- Engineering drawings often give angles in decimal degrees, while older survey plans use DMS or gradians.
Common angle conversions
| From | To |
|---|---|
| 1 ° | 0.01745329252 rad |
| 1 rad | 57.29577951 ° |
| 30 ° | 0.5235987756 rad |
| 45 ° | 0.7853981634 rad |
| 90 ° | 1.570796327 rad |
| 180 ° | 3.141592654 rad |
| 360 ° | 6.283185307 rad |
| 90 ° | 100 grad |
| 1 ° | 60 ′ |
| 1 ° | 17.77777778 mil |
| 1 mil | 0.9817477042 mrad |
| 1 mrad | 3.437746771 ′ |
Angle Converter FAQ
How do I convert degrees to radians?
Multiply by π and divide by 180. For example, 60° × π/180 = π/3 ≈ 1.047198 rad. The angle converter shows the formula for every pair.
How many degrees is 1 radian?
One radian is 180/π ≈ 57.29578 degrees — the angle at which the arc length equals the radius.
How do I convert decimal degrees to degrees, minutes and seconds?
Keep the whole degrees, multiply the decimal part by 60 for minutes, then multiply the leftover decimal by 60 for seconds. The DMS panel on this page does it both ways instantly.
What is a gradian?
A unit that divides a right angle into 100 parts and a full circle into 400. One gradian (also called a gon or grade) is 0.9°.
How many mils are in a circle?
NATO mils divide a circle into 6,400. A true milliradian divides it into 2,000π ≈ 6,283.2. The two are close but not the same.
What is an arcsecond?
One sixtieth of an arcminute, or 1/3,600 of a degree. Astronomers use arcseconds to measure tiny angles such as the apparent size of planets and the shift of nearby stars.