Radians to Milliradians (rad to mrad)

1 radian equals 1,000 milliradians exactly.

1 radian = 1,000 milliradians

How to convert radians to milliradians

To convert radians to milliradians, multiply the number of radians by 1,000. This factor is exact. To go the other way, multiply milliradians by 0.001.

milliradians = radians × 1,000

Example: 10 radians = 10,000 milliradians.

About these units

Radian (rad)

Metric (SI)Current unit

The SI unit of angle: the angle at the center of a circle subtended by an arc equal in length to the radius. A full turn is 2π radians (about 6.2832).

The radian builds its unit out of the circle itself. Lay a length of string equal to the radius along the circumference, and the angle that arc makes at the center is one radian. The SI Brochure calls it the coherent unit of plane angle, and it also serves as the coherent unit of phase angle, which grows by 2π radians over one period of a repeating wave. The same document notes that you could write rad as m/m, but warns that this is not intrinsic and can mislead, since angle is not the same kind of quantity as an ordinary ratio of lengths. It also records an alternative definition: a right angle equals π/2 radian.

A circle's circumference is 2π times its radius, so a full turn is 2π radians, about 6.2832, which makes one radian roughly 57.3 degrees, or about 0.159 of a turn. The SI Brochure's table of derived units expresses angular velocity in radians per second.

1 radian = 0.1591549431 turns (approximately).

Sources: BIPM SI Brochure, 9th edition (2019); Wikipedia: Angle; NIST Guide to the SI, Appendix B.8 (angle conversion factors)

Milliradian (mrad)

ScientificCurrent unit

One thousandth of a radian. Used in shooting and optics: 1 mrad subtends about 1 meter at a distance of 1 kilometer.

The milliradian cuts the radian into thousandths so that small angles turn into easy mental arithmetic. One mrad is about 0.0573 degree, or roughly 3.44 minutes of arc, and it covers about 1 meter at 1,000 meters, or 1 millimeter at 1 meter. That makes range estimation simple: the distance in meters is the target's size in millimeters divided by its angle in milliradians.

Swiss engineer Charles-Marc Dapples, a professor at the University of Lausanne, is credited with first using the unit in the mid-19th century. Today rifle scope reticles and many adjustment turrets are calibrated in it, and a 0.1 mrad click shifts the point of impact about 1 centimeter at 100 meters. A true milliradian gives 2000π, about 6,283.185, per turn, which is why military mils are rounded relatives rather than exact equivalents. Combining milliradian reticles with minute-of-arc turrets is generally discouraged.

1 milliradian = 0.0001591549431 turns (approximately).

Sources: Wikipedia: Milliradian; Wikipedia: Minute and second of arc; BIPM SI Brochure, 9th edition (2019)

Related conversions

See all angle units.