Not to be confused with Minute of arc.
The PSO-1 reticle in a Dragunov sniper rifle has 10 horizontal lines with 1-mil spacing, which can be used to compensate for wind drift or for range estimation.
Unit information
Unit system SI derived unit
Unit of Angle
Named after The metric prefix mille, Latin for "one thousand"
In units Dimensionless with an arc length one thousandth of the radius, i.e. 1 mm/m or 1 m/km
Unit conversions
1 mil in ... ... is equal to ...
turns    1/2000π turn
gons    ≈ 0.063662g
degrees    ≈ 0.057296°
arcminutes    ≈ 3.4377′

A milliradian, often called a mil or mrad, is an SI derived unit for angular measurement which is defined as a thousandth of a radian (0.001 radian). Mils are used in adjustment of firearm sights by adjusting the angle of the sight compared to the barrel (up, down, left or right). Mils are also used for comparing shot groupings, or to compare the difficulty of hitting different sized targets at different distances. Using optics with mil markings in the reticle one can make a range estimation of a known size target, or vice versa to determine a target size if the distance is known, a practice called "milling".

Milliradians are generally used for very small angles, which allows for precise mathematical simplifications to more easily calculate back and forth between the angular separation observed in an optic, linear subtension on target and range. In such applications it is useful to use a unit for target size that is a thousandth of the unit for range, for instance by using the metric units millimeters for target size and meters for range. This coincides with the definition of the milliradian where the arc length is defined as 1/1000 of the radius. A common adjustment value in firearm sights is 1 cm at 100 meters which equals 10 mm/100 m = 1/10 mil.

The true definition of a milliradian is based on a unit circle with a radius of one and an arc divided into 1000 mils per radian, hence 2000π or approximately 6283.185 milliradians in one turn. There are other definitions used for land mapping and artillery which are rounded to more easily be divided into smaller parts. For instance there are compasses with 6400 NATO mils, 6000 Warsaw Pact mils or 6300 Swedish "strecks" per turn instead of 360° or 2000π, achieving higher resolution than a 360° compass while also being easier to divide into parts than if true milliradians were used.

## History

The Palais de Rumine, one of the former buildings of the University of Lausanne.

The milliradian (approximately 6283.185 in a circle) was first used in the mid nineteenth century by Charles-Marc Dapples (1837–1920), a Swiss engineer and professor at the University of Lausanne.[1] 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. However, Imperial Russia used a different approach, dividing a circle into equilateral triangles (60° per triangle, 6 triangles in a circle)[citation needed] and hence 600 units to a circle.

Around the time of the start of World War I, France was experimenting with the use of milliemes (6400 in a circle) for use with artillery sights instead of decigrades (4000 in a circle). The United Kingdom was also trialing them to replace degrees and minutes. They were adopted by France although decigrades also remained in use throughout World War I. Other nations also used decigrades. The United States, which copied many French artillery practices, adopted mils (6400 in a circle). Before 2007 the Swedish defence forces used "streck" (6300 in a circle, streck meaning lines or marks) (together with degrees for some navigation) which is closer to the milliradian but then changed to NATO mils. After the Bolshevik Revolution and the adoption of the metric system of measurement (e.g. artillery replaced "units of base" with meters) the Red Army expanded the 600 unit circle into a 6000 mil one. Hence the Russian mil has a somewhat different origin than those derived from French artillery practices.

In the 1950s, NATO adopted metric units of measurement for land and general use. Mils, meters, and kilograms became standard, although degrees remained in use for naval and air purposes, reflecting civil practices.

## Mathematical principle

Left: An angle of 1 radian (marked green, approximately 57.3°) corresponds to an angle where the length of the arc (blue) is equal to the radius of the circle (red).
Right: A milliradian corresponds to 1/1000 of the angle of a radian. (The image on the right is exaggerated for illustration, as a milliradian is much smaller in reality).
For small observed angles (green) the arc length (blue) approaches the subtension (orange).

Use of the milliradian is practical because it is concerned with small angles, and when using radians the small angle approximation shows that the angle approximates to the sine of the angle, that is ${\displaystyle \sin \theta \simeq \theta }$. This allows a user to dispense with trigonometry and use simple ratios to determine size and distance with high accuracy for rifle and short distance artillery calculations by using the handy property of subtension: One mil approximately subtends one meter at a distance of one thousand meters.

More in detail, because ${\displaystyle {\text{subtension}}\simeq {\text{arc length}}}$, instead of finding the angular distance denoted by θ (Greek letter theta) by using the tangent function

${\displaystyle \theta _{\text{trig}}=\arctan {\frac {\text{subtension}}{\text{range}}}}$,

one can instead make a good approximation by using the definition of a radian and the simplified formula:

${\displaystyle \theta _{\text{rad}}={\frac {\text{subtension}}{\text{range}}}}$

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. Just like the radian, the milliradian is dimensionless, but unlike the radian where the same unit must be used for radius and arc length, the milliradian needs to have a ratio between the units where the subtension is a thousandth of the radius when using the simplified formula.

### Approximation error

The approximation error by using the simplified linear formula will increase as the angle increases. For example a

• 3.3 × 10-7 % (or 0.00000033%) error for an angle of 0.1 mil, for instance by assuming 0.1 mil equals 1 cm at 100 m[2]
• 0.03 % error for 30 mils, i.e assuming 30 mils equals 30 m at 1000 m[3]
• 2.9 % error for 300 mils, i.e. assuming 300 mils equals 300 m at 1000 m[4]

The approximation using mils is more precise than using another common system where 1′ is approximated as 1 inch and 100 yards, where comparably there is a:

• 4.5 % error by assuming that an angle of 1' equals 1 in at 100 yd[5]
• 55 % error for 100', i.e. assuming 100' equals 100 in at 100 yd[6]
• 953 % error for 1000', i.e. assuming 1000' equals 1000 in at 100 yd[7]

Mil 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. New shooters are often explained the principle of subtensions in order to understand that a milliradian is an angular measurement. Subtension is the physical amount of space covered by an angle and varies with distance. Thus, the subtension corresponding to a mil (either in a mil reticle or in mil adjustments) varies with range. Knowing subtensions at different ranges can be useful for sighting in a firearm if there is no optic with a mil reticle available, but involve mathematical calculations and are therefore not used very much in practical applications. Subtensions always change with distance, but a mil (as observed through an optic) is always a mil regardless of distance. Therefore ballistic tables and shot corrections are given in mils thereby avoiding the need for mathematical calculations.

If a rifle scope has mil markings in the reticle (or there is a spotting scope with a mil reticle available), the reticle can be used to measure how many mils to correct a shot even without knowing the shooting distance. For instance, assuming a precise shot fired by an experienced shooter missed the target by 0.8 mils as seen through an optic, and the firearm sight has 0.1 mil adjustments, the shooter must then dial 8 clicks on the scope to hit the same target under the same conditions.

### Common click values

General purpose scopes

1/4′, 0.1 mil and 1/2′ is used in general purpose sights for hunting, target and long range shooting at varied distances, since the click values are fine enough to get dialed in for most target shooting and coarse enough to keep the number of clicks down when dialing.

Speciality scopes

0.025 mil, 1/8′ and 0.05 mil is used in speciality sights for extreme precision at fixed target ranges, such as ISSF 10 to 300 meter rifle and benchrest shooting. The small adjustment value means that the line of sight can be adjusted in very small increments. These fine adjustments are however not very well suited for dialing between varied distances such as in field shooting because of the high number of clicks that will be required to move the line of sight, making it easier to lose track of the number of clicks than in scopes with larger click adjustments. For instance to move the line of sight 0.4 mil, a 0.1 mil scope must be adjusted 4 clicks, while comparably a 0.05 mil and 0.025 mil scope must be adjusted 8 and 16 clicks respectively.

Other

0.15 mil and 0.2 mil clicks can be found in some short range sights, mostly with capped turrets, but are not very widely used.

### Subtensions at different distances

Subtensions in mil based optics are particularly useful together with target sizes and shooting distances in metric units. Since a mil is an angular measurement, the length of the area covered by the angle increase with distance. A common scope adjustment increment in rifle scopes is 0.1 mil, which are sometimes called "one centimeter clicks" since 0.1 mil equals exactly 1 cm at 100 meters, 2 cm at 200 meters, etc. Similarly, an adjustment click on a scope with 0.2 mil adjustment will move the point of bullet impact 2 cm at 100 m and 4 cm at 200 m, etc. Impact change at different ranges can be calculated by the formula:

${\displaystyle {\text{adjustment in mils}}={\frac {\text{subtension in mm}}{\text{range in m}}}}$

For instance:

• 20 mm/50 m = 0.4 mils, or 4 clicks with a 1/10 mil adjustment scope.
• 50 mm/1000 m = 0.05 mils, or 1 click with a 0.05 mil adjustment scope.

In firearm optics, where 0.1 mil per click is the most common mil based adjustment value, another common way to think is that:

One click changes the impact as many centimeters as there are hundreds of meters.

I.e 1 cm at 100 meters, 2.25 cm at 225 meters, 0.5 cm at 50 meters, etc. Some common scope adjustment values in mils (converted to centimeters at hundreds of meters) can be seen in the table below.

Angle @ 100 m @ 200 m @ 300 m @ 400 m @ 500 m @ 600 m @ 700 m @ 800 m @ 900 m @ 1000 m
0.1 mil 1 cm 2 cm 3 cm 4 cm 5 cm 6 cm 7 cm 8 cm 9 cm 10 cm
0.15 mil 1.5 cm 3 cm 4.5 cm 6 cm 7.5 cm 9 cm 10.5 cm 12 cm 13.5 cm 15 cm
0.2 mil 2 cm 4 cm 6 cm 8 cm 10 cm 12 cm 14 cm 16 cm 18 cm 20 cm

### Adjustment range and base tilt

The horizontal and vertical adjustment range of a firearm sight is often advertised by the manufacturer using mils. For instance a rifle scope may be advertised as having a vertical adjustment range of 20 mils, which means that by turning the turret the bullet impact can be moved a total of 20 meters at 1000 meters (or 2 m at 100 m, 4 m at 200 m, 6 m at 300 m etc.). The horizontal and vertical adjustment ranges can be different for a particular sight, for instance a scope may have 20 mils vertical and 10 mils horizontal adjustment. Elevation differ between models, but about 10–11 mils are common in hunting scopes, while scopes made for long range shooting usually can have an adjustment range of 30–50 mils.[citation needed]

Sights can either be mounted in neutral or tilted mounts. In a neutral mount (also known as "flat base" or non-tilted mount) the sight will point reasonably parallel to the barrel, and be close to a zero at 100 meters (about 1 mil low depending on rifle and caliber). After zeroing at 100 meters the sight will thereafter always have to be adjusted upwards to compensate for bullet drop at longer ranges, and therefore the adjustment below zero will never be used. This means that when using a neutral mount only about half of the scope's total elevation will be usable for shooting at longer ranges:

${\displaystyle {\text{usable elevation in neutral mount}}={\frac {\text{scope's total elevation}}{2}}}$

In most regular sport and hunting rifles (except for in long range shooting), sights are usually mounted in neutral mounts. This is done because the optical quality of the scope is best in the middle of its adjustment range, and only being able to use half of the adjustment range to compensate for bullet drop is seldom a problem at short and medium range shooting.

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:

• 3 mils, which equals 3 m at 1000 m (or 0.3 m at 100 m)
• 6 mils, which equals 6 m at 1000 m (or 0.6 m at 100 m)
• 9 mils, which equals 9 m at 1000 m (or 0.9 m at 100 m)

With a tilted mount the maximum usable scope elevation can be found by:

${\displaystyle {\text{maximum elevation with tilted mount}}={\frac {\text{scope's total elevation}}{2}}+{\text{base tilt}}}$

The adjustment range needed to shoot at a certain distance vary with firearm, caliber and load. For example, with a certain .308 load and firearm combination, the bullet may drop 13 mils at 1000 meters (13 meters). To be able to reach out, one could either:

• Use a scope with 26 mils of adjustment in a neutral mount, to get a usable adjustment of 26 mils/2 = 13 mils
• Use a scope with 14 mils of adjustment and a 6 mil tilted mount to achieve a maximum adjustment of 14 mils/2 + 6 = 13 mils

## Shot groupings

Main article: Shot grouping
Two shot groupings obtained with different firearms at different distances.
The group on the left measures about 13 mm and was fired at about 45 meters, which equals 13 mm/45 m = 0.289 mils.
The group on the right measures about 7 mm and was fired at about 90 meters, which equals 7 mm/90 m = 0.078 mils.

A shot grouping is the spread of multiple shots on a target, taken in one shooting session. The group size on target in milliradians can be obtained by measuring the spread of the rounds on target in millimeters with a caliper and dividing by the shooting distance in meters. This way, using mils, one can easily compare shot groupings or target difficulties at different shooting distances.

${\displaystyle {\text{group size in mils}}={\frac {\text{group size in mm}}{\text{range in m}}}}$

If the firearm is attached in a fixed mount and aimed at a target, the shot grouping measures the firearms mechanical precision and the uniformity of the ammunition. When the firearm also is held by a shooter, the shot grouping partly measures the precision of the firearm and ammunition, and partly the shooter's consistency and skill. 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.

## Range estimation with mil reticles

"FinDot" reticle as used by Finnish Defence Forces snipers (a regular Mil-dot reticle with the addition of 400 m – 1200 m holdover (stadiametric) rangefinding brackets for 1 meter high or 0.5 meter wide targets at 400, 600, 800, 1000 and 1200 m).

Many telescopic sights used on rifles have reticles that are marked in mils. This can either be accomplished with lines or dots, and the latter is generally called mil-dots. The mil reticle serves two purposes, range estimation and trajectory correction.

With a mil reticle-equipped scope the distance to an object can be estimated with a fair degree of accuracy by a trained user by determining how many angular mils an object of known size subtends. Once the distance is known, the drop of the bullet at that range (see external ballistics), converted back into angular mils, can be used to adjust the aiming point. Generally mil-reticle scopes have both horizontal and vertical crosshairs marked; the horizontal and vertical marks are used for range estimation and the vertical marks for bullet drop compensation. Trained users, however, can also use the horizontal dots to compensate for bullet drift due to wind. Mil-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.

Estimating mils with hands
Mildot chart as used by snipers

Angle can be used for either calculating target size or range if one of them are known. Where the range is known the angle will give the size, where the size is known then the range is given. When out in the field angle can be measured approximately by using calibrated optics or roughly using one's fingers and hands. With an outstretched arm one finger is approximately 30 mils wide, a fist 150 mils and a spread hand 300 mils.

Mil reticles often have dots or marks with a spacing of one mil in between, but graduations can also be finer and coarser (i.e. 0.8 or 1.2 mil).

### Units for target size and range

While a radian is defined as an angle on the unit circle where the arc and radius have equal length, a milliradian is defined as the angle where the arc length is one thousandth of the radius. Therefore, when using milliradians for range estimation, the unit used for target distance needs to be thousand times as large as the unit used for target size. Metric units are particularly useful in conjunction with a mil reticle because the mental arithmetic is much simpler with decimal units, thereby requiring less mental calculation in the field. Using the range estimation formula with the units meters for range and millimeters for target size it is just a matter of moving decimals and do the division, without the need of multiplication with additional constants, thus producing fewer rounding errors.

${\displaystyle {\text{distance in meters}}={\frac {\text{target in millimeters}}{\text{angle in mils}}}}$

The same holds true for calculating target distance in kilometers using target size in meters.

${\displaystyle {\text{distance in kilometers}}={\frac {\text{target in meters}}{\text{angle in mils}}}}$

If using the imperial units yards for distance and inches for target size, one has to multiply by a factor of 100036 ≈ 27.78, since there are 36 inches in one yard.

${\displaystyle {\text{distance in yards}}={\frac {\text{target in inches}}{\text{angle in mils}}}\times 27.78}$

Also, in general the same unit can be used for subtension and range if multiplied with a factor of thousand, i.e.

${\displaystyle {\text{distance in meters}}={\frac {\text{target in meters}}{\text{angle in mils}}}\times 1,000}$

### Practical examples

Land Rovers are about 3 to 4 m long, "smaller tank" or APC/MICV at about 6 m (e.g. T-34 or BMP) and about 10 m for a "big tank." From the front a Land Rover is about 1.5 m, most tanks around 3 - 3.5 m. So a SWB Land Rover from the side are 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 wide (1500 mm) is measured to 2.8 mils in the reticle, the range can be estimated to:

${\displaystyle {\text{distance in meters}}={\frac {1500~{\text{mm}}}{2.8~{\text{mils}}}}=535.7~{\text{m}}}$

So if the above-mentioned 6 m long BMP (6000 mm) is viewed at 6 mils its distance is 1000 m, and if the angle of view is twice as large (12 mils) 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:

${\displaystyle {\text{distance in meters}}={\frac {\text{size in mm}}{\text{angle in mils}}}\times {\frac {\text{mag}}{10}}}$

Where mag is scope magnification. However, a user should verify this with their individual scope since some are not calibrated at 10×. As above target distance and target size can be given in any two units of length with a ratio of 1000:1.

## Mixing mil and minutes of arc

It is possible to purchase rifle scopes with a mix of for instance a mil reticle and minute-of-arc turrets (or vice versa), but it is general consensus that such mixing should be avoided. It is preferred to either have both a mil reticle and mil adjustment (mil/ mil), or a minute-of-arc reticle and minute-of-arc adjustment to utilize the strength of each system. Then the shooter can know exactly how many clicks to correct based on what he sees in the reticle.

If using a mixed system scope that has a mil reticle and arcminute adjustment, one way to make use of the reticle for shot corrections is to exploit that 14′ approximately eqauals 4 mils, and thereby multiplying an observed corrections in mils by a fraction of 144 when adjusting the turrets.

### Conversion table

In the table below conversions from mil to metric values are exact (e.g. 0.1 mil equals exactly 1 cm at 100 meters), while conversions of minutes of arc to both metric and imperial values are approximate.

Minute of arc equivalent
(decimal)
Mil equivalent mm @ 100 m cm @ 100 m in @ 100 m in @ 100 y
0.025 mil 0.086′ 0.025 mil 2.5 mm 0.25 cm 0.0985 in 0.09 in
1/8′ 0.125′ 0.036 mil 3.64 mm 0.36 cm 0.14 in 0.13 in
0.05 mil 0.172′ 0.05 mil 5 mm 0.5 cm 0.197 in 0.18 in
1/4′ 0.25′ 0.073 mil 7.27 mm 0.73 cm 0.29 in 0.26 in
0.1 mil 0.344′ 0.1 mil 10 mm 1 cm 0.39 in 0.36 in
1/2′ 0.5′ 0.145 mil 14.54 mm 1.45 cm 0.57 in 0.52 in
0.15 mil 0.516′ 0.15 mil 15 mm 1.5 cm 0.59 in 0.54 in
0.2 mil 0.688′ 0.2 mil 20 mm 2 cm 0.79 in 0.72 in
1′ 1.0′ 0.291 mil 29.1 mm 2.91 cm 1.15 in 1.047 in
1 mil 3.438′ 1 mil 100 mm 10 cm 3.9 in 3.6 in
• 0.1 mil equals exactly 1 cm at 100 m
• 1 mil ≈ 3.44′, so 1/10 mil ≈ 1/3
• 1′ ≈ 0.291 mil (or 2.91 cm at 100 m, approximately 3 cm at 100 m)

## Definitions for maps and artillery

Map measure M/70 of the NATO member Denmark with the full circle divided into 6400 NATO mils
In the Swiss Army, 6400 "artillery per milles" ("Artilleriepromille") are used to indicate an absolute indication of direction by using the notation that 0 A ‰ (corresponding to 6400 A ‰) points to the north, instead of using NATO mils where direction is always relative to the target (0 or 6400 NATO mils is always towards target).

Because of the definition of pi, in a circle with a diameter of one there are 2000π milliradians (≈ 6283.185 mil) per full turn. 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.[citation needed]

For maps and artillery, three rounded definitions are used which are close to the real definition, but more easily can be divided into parts. The different map and artillery definitions are:

• 1/6400 of a circle in NATO countries.
• 1/6000 of a circle in the former Soviet Union and Finland (Finland phasing out the standard in favour of the NATO standard).
• 1/6300 of a circle in Sweden. The Swedish term for this is streck, literally "line". Sweden (and Finland) have not been part of NATO nor the Warsaw Pact. Note however that Sweden has changed its map grid systems and angular measurement to those used by NATO, so the "streck" measurement is obsolete.

Reticles in some artillery sights are calibrated to the relevant artillery definition for that military, i.e. the Carl Zeiss OEM-2 artillery sight made in DDR from 1969 to 1976 is calibrated for the eastern block 6000 mil circle.[citation needed]

### Conversion table

Conversions between units
Milliradian NATO mil Warsaw Pact Mil Swedish streck Degrees Minute of arc
1 milliradian = 1 1.018592 0.954930 1.002677 0.057296 3.437747
1 NATO mil = 0.981719 1 0.9375 0.984375 0.05625 3.375
1 Warsaw Pact mil = 1.047167 1.066667 1 1.05 0.06 3.6
1 Swedish streck = 0.997302 1.015873 0.952381 1 0.057143 3.428572
1 degree = 17.452778 17.777778 16.666667 17.5 1 60
1 minute of arc = 0.290880 0.296297 0.277778 0.291667 0.016667 1

(Values in bold face are exact.)

• 1 trigonometric milliradian (mil) ≈ 3.43774677078493′
• 1 NATO mil = 3.375′ exactly
• 1 Warsaw Pact mil = 3.6′ exactly

### Use in artillery sights

Artillery uses angular measurement in gun laying, the azimuth between the gun and its target many kilometres away and the elevation angle of the barrel. This means that artillery uses mils to graduate indirect fire azimuth sights (called dial sights or panoramic telescopes), their associated instruments (directors or aiming circles), their elevation sights (clinometers or quadrants), together with their manual plotting devices, firing tables and fire control computers.

Artillery spotters typically use their calibrated binoculars to walk fire 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 metres.