An angle measures rotation, and different trades divide a full circle in different ways. Everyday geometry, navigation and carpentry use 360 degrees; mathematics, physics and programming languages use radians; surveyors in parts of Europe use 400 gradians; astronomers count arcseconds; artillery and long-range shooters use mils or milliradians. This converter translates between 11 angle units and shows the same angle in degrees-minutes-seconds form, as a multiple of π and normalized to the 0–360° range.
How to use the angle converter
- Enter the angle in Value. Negative angles and angles over a full turn are accepted.
- Choose its unit in From and the target unit in To.
- The tape shows the result, the factors, the DMS form, the multiple of π and the normalized angle.
- The table lists the angle in every supported unit.
Angle conversion formula
The radian is the SI unit. One radian is the angle at which the arc length equals the radius, so a full circle is 2π radians. Every other unit is a fraction of that:
The two most-used conversions are
Worked example: radians from a programming function
A program returns an angle of 2.5 radians, and you want degrees for a drawing:
2.5 × 180 ÷ π = 143.2394°
In DMS: 0.2394488° × 60 = 14.367′, and 0.367′ × 60 = 22.02″, so 143° 14′ 22.02″.
As a fraction of a full turn: 143.2394 ÷ 360 = 0.3979 turn.
The reverse is just as common. A 45° angle is 45 × π/180 = π/4 ≈ 0.785398 rad, which is the value to pass to a sine or cosine function that expects radians.
Angle units compared
| Unit | Per full circle | Degrees per unit | Typical use |
|---|---|---|---|
| Degree (°) | 360 | 1 | Geometry, navigation, construction |
| Radian (rad) | 2π ≈ 6.2832 | 57.29578 | Math, physics, programming |
| Gradian (grad, gon) | 400 | 0.9 | European surveying |
| Arcminute (′) | 21,600 | 1/60 | Astronomy, maps, optics |
| Arcsecond (″) | 1,296,000 | 1/3,600 | Astronomy, geodesy |
| Turn (revolution) | 1 | 360 | Rotating machinery |
| Milliradian (mrad) | ≈ 6,283.2 | 0.0572958 | Optics, ballistics |
| NATO mil | 6,400 | 0.05625 | Military compasses, artillery |
| Hour angle | 24 | 15 | Astronomy (right ascension) |
| Compass point | 32 | 11.25 | Traditional marine navigation |
Useful angle facts
Small-angle shortcuts. For angles under about 10°, the angle in radians is close to its sine and tangent. That is why one milliradian subtends almost exactly 1 m at 1,000 m (or 10 cm at 100 m), and one arcminute subtends about 1.047 inches at 100 yards, the “minute of angle” rule shooters use.
Negative and large angles. Rotation counterclockwise is positive by convention in mathematics; clockwise is negative. An angle of −90° points the same way as 270°, and 450° points the same way as 90°. The normalized value on the tape removes whole turns so you can see the direction.
Calculator modes. Most trigonometry mistakes come from a calculator in the wrong mode. If sin(30) returns −0.988 instead of 0.5, the calculator is in radian mode. Converting to radians first, or switching to DEG, fixes it.
Bearings versus math angles. Compass bearings start at north and run clockwise, while mathematical angles start at the positive x-axis and run counterclockwise. Converting between them is a shift and a sign change, not a unit conversion.
For latitude and longitude, use the decimal degrees to DMS converter, which adds N/S/E/W labels. To evaluate trig functions, see the trig functions calculator, or explore exact values on the unit circle calculator.
Frequently asked questions
How do I convert degrees to radians?
Multiply by π/180, about 0.0174533. For example, 45° × π/180 = π/4 ≈ 0.785398 rad. To go from radians to degrees, multiply by 180/π, about 57.2958.
What is a gradian?
A gradian (also called a gon or grad) is 1/400 of a full circle, so a right angle is exactly 100 gradians. It is used in some surveying work in Europe and appears as the GRAD mode on scientific calculators.
What is the difference between a milliradian and a mil?
A true milliradian is 1/1,000 radian, giving about 6,283 per circle. The NATO mil used on military compasses rounds that to exactly 6,400 per circle, making it 0.05625°, about 1.8% smaller than a milliradian.
How many arcseconds are in a degree?
3,600. A degree has 60 arcminutes and each arcminute has 60 arcseconds. Astronomers and surveyors use these units for very small angles.