Milliradians are generally used for very small angles, which allows for very accurate mathematical approximations to more easily calculate with direct proportions, back and forth between the angular separation observed in an optic, linear subtension on target, and range.

[1] There are also other definitions used for land mapping and artillery which are rounded to more easily be divided into smaller parts for use with compasses, which are then often referred to as "mils", "lines", or similar.

For instance there are artillery sights and compasses with 6,400 NATO mils, 6,000 Warsaw Pact mils or 6,300 Swedish "streck" per turn instead of 360° or 2π radians, achieving higher resolution than a 360° compass while also being easier to divide into parts than if true milliradians were used.

The milliradian (approximately 6,283.185 in a circle) was first used in the mid-19th century by Charles-Marc Dapples (1837–1920), a Swiss engineer and professor at the University of Lausanne.

[2] Degrees and minutes were the usual units of angular measurement but others were being proposed, with "grads" (400 gradians in a circle) under various names having considerable popularity in much of northern Europe.

NATO mils, meters, and kilograms became standard, although degrees remained in use for naval and air purposes, reflecting civil practices.

, instead of finding the angular distance denoted by θ (Greek letter theta) by using the tangent function one can instead make a good approximation by using the definition of a radian and the simplified formula: Since a radian is mathematically defined as the angle formed when the length of a circular arc equals the radius of the circle, a milliradian, is the angle formed when the length of a circular arc equals ⁠1/1000⁠ of the radius of the circle.

For example, a The approximation using mrad is more precise than using another common system where 1′ (minute of arc) is approximated as 1 inch at 100 yards, where comparably there is a: where Milliradian adjustment is commonly used as a unit for clicks in the mechanical adjustment knobs (turrets) of iron and scope sights both in the military and civilian shooting sports.

Knowing subtensions at different ranges can be useful for sighting in a firearm if there is no optic with an mrad reticle available, but involves mathematical calculations, and is therefore not used very much in practical applications.

For instance, assuming a precise shot fired by an experienced shooter missed the target by 0.8 mrad as seen through an optic, and the firearm sight has 0.1 mrad adjustments, the shooter must then dial 8 clicks on the scope to hit the same target under the same conditions.

Subtensions in mrad based optics are particularly useful together with target sizes and shooting distances in metric units.

See the table below The horizontal and vertical adjustment range of a firearm sight is often advertised by the manufacturer using mrads.

However, in long range shooting tilted scope mounts are common since it is very important to have enough vertical adjustment to compensate for the bullet drop at longer distances.

For this purpose scope mounts are sold with varying degrees of tilt, but some common values are: With a tilted mount the maximum usable scope elevation can be found by: The adjustment range needed to shoot at a certain distance varies with firearm, caliber and load.

This way, using milliradians, one can easily compare shot groupings or target difficulties at different shooting distances.

Often the shooters' skill is the most important element towards achieving a tight shot grouping,[citation needed] especially when competitors are using the same match grade firearms and ammunition.

Once the distance is known, the drop of the bullet at that range (see external ballistics), converted back into milliradians, can be used to adjust the aiming point.

Milliradian-reticle-equipped scopes are well suited for long shots under uncertain conditions, such as those encountered by military and law enforcement snipers, varmint hunters and other field shooters.

These riflemen must be able to aim at varying targets at unknown (sometimes long) distances, so accurate compensation for bullet drop is required.

From the front a Land Rover is about 1.5 m wide, most tanks around 3–3.5 m. So a SWB Land Rover from the side is one finger wide at about 100 m. A modern tank would have to be at a bit over 300 m. If, for instance a target known to be 1.5 m in height (1500 mm) is measured to 2.8 mrad in the reticle, the range can be estimated to: So if the above-mentioned 6 m long BMP (6000 mm) is viewed at 6 mrad its distance is 1000 m, and if the angle of view is twice as large (12 mrad) the distance is half as much, 500 m. When used with some riflescopes of variable objective magnification and fixed reticle magnification (where the reticle is in the second focal plane), the formula can be modified to: Where mag is scope magnification.

It is possible to purchase rifle scopes with a mrad reticle and minute-of-arc turrets, but it is general consensus that such mixing should be avoided.

In other words, one real milliradian covers just under ⁠1/6283⁠ of the circumference of a circle, which is the definition used by telescopic rifle sight manufacturers in reticles for stadiametric rangefinding.

Artillery spotters typically use their calibrated binoculars to move fired projectiles' impact onto a target.

Here they know the approximate range to the target and so can read off the angle (+ quick calculation) to give the left/right corrections in meters.

The milliradian (and other SI multiples) is also used in other fields of science and technology for describing small angles, i.e. measuring alignment,[12][13] collimation,[14] and beam divergence in optics,[15] and accelerometers and gyroscopes in inertial navigation systems.