Angle Converter
Convert between Angle units. Choose from 11 different units including Degrees, Radians, Gradians.
Popular Conversions
All Units
| From | Degrees (°) |
|---|---|
| Degrees (°) | 1 |
| Radians (rad) | 57.29577951 |
| Gradians (gon) | 0.9 |
| Arcminutes (′) | 0.016666667 |
| Arcseconds (″) | 0.000277778 |
| Turns (tr) | 360 |
| Milliradians (mrad) | 0.057295779 |
| Signs (sign) | 30 |
| NATO Mils (mil) | 0.05625 |
| Quadrants (quad) | 90 |
| Sextants (sxt) | 60 |
Slicing a circle
An angle is a measure of rotation or separation between two rays, and the units used to describe it are essentially different ways of slicing the same circle. A degree carves it into 360 wedges, a gradian into 400, a NATO mil into 6400, and a radian — the odd one out — defines the angle whose arc length equals the radius, giving exactly 2π radians per turn.
A Babylonian inheritance
The 360-degree circle traces back to Babylonian base-60 arithmetic, a system so well suited to fractions that it survived into modern timekeeping and navigation. Arcminutes (1/60°) and arcseconds (1/3600°) preserve that sexagesimal logic to this day. The radian, by contrast, is a child of 18th-century calculus — natural because it makes the derivative of sine equal cosine without any awkward conversion factor. The gradian was a French Revolutionary push toward decimalization that mostly failed, except in surveying and parts of continental cadastral work.
Where each unit shows up today
Degrees dominate everyday life, navigation, and machine controls. Radians dominate physics, calculus, and any code that calls a trig function — sin(π) works precisely because radians are dimensionless ratios. Astronomers still talk in arcseconds when describing stellar parallax or telescope resolution, GPS coordinates use degree-minute-second notation, and milliradians (mrad) are standard on rifle scopes and laser optics.
The mil trap and other snares
The single most common error is conflating a milliradian with a NATO mil. A true mrad gives 6283.19 per circle; a NATO artillery mil gives exactly 6400, rounded for mental arithmetic so that one mil subtends roughly one meter at one kilometer. Scopes labeled "mil-dot" almost always mean mrad, not NATO mils, despite the shared nickname. Two more pitfalls: forgetting that calculators in degree mode silently mangle radian-based formulas, and assuming gradians and degrees are interchangeable when reading European survey plans, where 100 gons (not 90 degrees) marks a right angle.