A milliradian (
SI-symbol mrad, sometimes also abbreviated mil) is an
SI derived unit
SI derived units are units of measurement derived from the
seven SI base units specified by the International System of Units (SI). They can be expressed as a product (or ratio) of one or more of the base units, possibly scaled by an appropriat ...
for
angular measurement
In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight lines at a point. Formally, an angle is a figure lying in a plane formed by two rays, called the '' sides'' of the angle, sharing ...
which is defined as a thousandth of a
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at ...
(0.001 radian). Milliradians are used in adjustment of
firearm
A firearm is any type of gun that uses an explosive charge and is designed to be readily carried and operated by an individual. The term is legally defined further in different countries (see legal definitions).
The first firearms originate ...
sights by adjusting the angle of the sight compared to the barrel (up, down, left, or right). Milliradians are also used for comparing
shot groupings, or to compare the difficulty of hitting different sized
shooting target
Shooting targets are objects in various forms and shapes that are used for pistol, rifle, shotgun and other shooting sports, as well as in darts, target archery, crossbow shooting and other non-firearm related sports. The center is often call ...
s at different distances. When using a scope with both mrad adjustment and a
reticle
A reticle or reticule, also known as a graticule or crosshair, is a pattern of fine lines or markings built into the eyepiece of an optical device such as a telescopic sight, spotting scope, theodolite, optical microscope or the electronic v ...
with mrad markings (called an "mrad/mrad scope"), the shooter can use the reticle as a
ruler
A ruler, sometimes called a rule, scale, line gauge, or metre/meter stick, is an instrument used to make length measurements, whereby a length is read from a series of markings called "rules" along an edge of the device. Usually, the instr ...
to count the number of mrads a shot was off-target, which directly translates to the sight adjustment needed to hit the target with a follow-up shot. Optics with mrad markings in the reticle can also be used to 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 very accurate mathematical approximations to more easily calculate with
direct proportions, back and forth between the
angular separation observed in an optic, linear
subtension
In geometry, an angle subtended (from Latin for "stretched under") by a line segment at an arbitrary vertex (geometry), vertex is formed by the two ray (geometry), rays between the vertex and each endpoint (geometry), endpoint of the segment.
...
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
millimeter
330px, Different lengths as in respect of the electromagnetic spectrum, measured by the metre and its derived scales. The microwave is between 1 metre to 1 millimetre.
The millimetre (American and British English spelling differences#-re, -er, i ...
s for target size and
meter
The metre (or meter in US spelling; symbol: m) is the base unit of length in the International System of Units (SI). Since 2019, the metre has been defined as the length of the path travelled by light in vacuum during a time interval of of ...
s for range. This coincides with the definition of the milliradian where the arc length is defined as of the radius. A common adjustment value in firearm sights is 1 cm at 100 meters which equals = mrad.
The true definition of a milliradian is based on a
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
with a radius of
one
1 (one, unit, unity) is a number, numeral, and glyph. It is the first and smallest positive integer of the infinite sequence of natural numbers. This fundamental property has led to its unique uses in other fields, ranging from science to sp ...
and an
arc divided into 1,000 mrad per radian, hence 2,000
π or approximately 6,283.185 milliradians in one
turn, and rifle scope adjustments and reticles are calibrated to this definition. There are also other definitions used for
land mapping and
artillery
Artillery consists of ranged weapons that launch Ammunition, munitions far beyond the range and power of infantry firearms. Early artillery development focused on the ability to breach defensive walls and fortifications during sieges, and l ...
which are rounded to more easily be divided into smaller parts for use with
compass
A compass is a device that shows the cardinal directions used for navigation and geographic orientation. It commonly consists of a magnetized needle or other element, such as a compass card or compass rose, which can pivot to align itself with No ...
es, 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.
History

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
Swiss most commonly refers to:
* the adjectival form of Switzerland
* Swiss people
Swiss may also refer to: Places
* Swiss, Missouri
* Swiss, North Carolina
* Swiss, West Virginia
* Swiss, Wisconsin
Other uses
* Swiss Café, an old café located ...
engineer and professor at the
University of Lausanne
The University of Lausanne (UNIL; ) in Lausanne, Switzerland, was founded in 1537 as a school of Protestant theology, before being made a university in 1890. The university is the second-oldest in Switzerland, and one of the oldest universities ...
. 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 triangle
An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
s (60° per triangle, 6 triangles in a circle) and hence 600 units to a circle.
Around the time of the start of
World War I
World War I or the First World War (28 July 1914 – 11 November 1918), also known as the Great War, was a World war, global conflict between two coalitions: the Allies of World War I, Allies (or Entente) and the Central Powers. Fighting to ...
, France was experimenting with the use of millièmes or angular mils (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 angular mils, later known as NATO mils. 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 circle. Hence the Russian mil has a somewhat different origin than those derived from French artillery practices.
In the 1950s,
NATO
The North Atlantic Treaty Organization (NATO ; , OTAN), also called the North Atlantic Alliance, is an intergovernmental organization, intergovernmental Transnationalism, transnational military alliance of 32 Member states of NATO, member s ...
adopted metric units of measurement for land and general use. NATO mils, meters, and kilograms became standard, although degrees remained in use for naval and air purposes, reflecting civil practices.
Mathematical principle
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
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite th ...
of the angle, that is
. This allows a user to dispense with
trigonometry
Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The fiel ...
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 mrad approximately subtends one meter at a distance of one thousand meters''.
More in detail, because
, instead of finding the
angular distance
Angular distance or angular separation is the measure of the angle between the orientation (geometry), orientation of two straight lines, ray (geometry), rays, or vector (geometry), vectors in three-dimensional space, or the central angle subtende ...
denoted by θ (Greek letter
theta
Theta (, ) uppercase Θ or ; lowercase θ or ; ''thē̂ta'' ; Modern: ''thī́ta'' ) is the eighth letter of the Greek alphabet, derived from the Phoenician letter Teth 𐤈. In the system of Greek numerals, it has a value of 9.
Gree ...
) by using the
tangent function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
:
,
one can instead make a
good approximation by using the definition of a radian and the simplified formula:
:
Since a
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at ...
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 of the radius of the circle. Just like the radian, the milliradian is
dimensionless
Dimensionless quantities, or quantities of dimension one, are quantities implicitly defined in a manner that prevents their aggregation into units of measurement. ISBN 978-92-822-2272-0. Typically expressed as ratios that align with another sy ...
, 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
* % (or parts per billion) error for an angle of 0.1 mrad, for instance by assuming 0.1 mrad equals 1 cm at 100 m
* 0.03% error for 30 mrad, i.e. assuming 30 mrad equals 30 m at 1 km
* 2.9% error for 300 mrad, i.e. assuming 300 mrad equals 300 m at 1 km
The approximation using mrad is more precise than using another common system where 1′ (
minute of arc
A minute of arc, arcminute (abbreviated as arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to of a degree. Since one degree is of a turn, or complete rotation, one arcminute is of a tu ...
) is approximated as 1 inch at 100 yards, where comparably there is a:
* 4.72% error by assuming that an angle of 1′ equals 1 inch at 100 yd
* 4.75% error for 100′, i.e. assuming 100′ equals 100 in at 100 yd
* 7.36% error for 1000′, i.e. assuming 1000′ equals 1000 inches at 100 yd
Sight adjustment
Milliradian adjustment is commonly used as a unit for clicks in the mechanical adjustment knobs (turrets) of
iron
Iron is a chemical element; it has symbol Fe () and atomic number 26. It is a metal that belongs to the first transition series and group 8 of the periodic table. It is, by mass, the most common element on Earth, forming much of Earth's o ...
and
scope sights both in the military and civilian
shooting sports
Shooting sports is a group of competitive sport, competitive and recreational sporting activities involving proficiency tests of accuracy, precision and speed in shooting — the art of using ranged weapons, mainly small arms (firearms and airg ...
. 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 mrad (either in an mrad reticle or in mrad adjustments) varies with range. 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. Subtensions always change with distance, but an mrad (as observed through an optic) is always an mrad regardless of distance. Therefore,
ballistic tables and shot corrections are given in mrads, thereby avoiding the need for mathematical calculations.
If a rifle scope has mrad markings in the reticle (or there is a
spotting scope with an mrad reticle available), the reticle can be used to measure how many mrads 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 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.
Common click values
; General purpose scopes: Gradations (clicks) of ′, mrad and ′ are used in general purpose sights for hunting, target and
long range shooting
Long range shooting is a collective term for shooting sport, shooting disciplines where the distance to the target is significant enough that the shooter has to put effort into calculating external ballistics, various ballistic factors, esp ...
at varied distances. 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: mrad, ′ and mrad are used in speciality scope sights for extreme precision at fixed target ranges such as
benchrest shooting. Some specialty iron sights used in
ISSF 10 m,
50 m and
300 meter rifle come with adjustments in either mrad or mrad. The small adjustment value means these sights 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 mrad, a 0.1 mrad scope must be adjusted 4 clicks, while comparably a 0.05 mrad and 0.025 mrad scope must be adjusted 8 and 16 clicks respectively.
; Others: mrad and mrad can be found in some short range sights, mostly with capped turrets, but are not very widely used.
Subtensions at different distances
Subtension refers to the length between two points on a target, and is usually given in either centimeters, millimeters or inches. Since an mrad is an angular measurement, the subtension covered by a given angle (
angular distance
Angular distance or angular separation is the measure of the angle between the orientation (geometry), orientation of two straight lines, ray (geometry), rays, or vector (geometry), vectors in three-dimensional space, or the central angle subtende ...
or
angular diameter
The angular diameter, angular size, apparent diameter, or apparent size is an angular separation (in units of angle) describing how large a sphere or circle appears from a given point of view. In the vision sciences, it is called the ''visual an ...
) increases with viewing distance to the target. For instance the same angle of 0.1 mrad will subtend 10 mm at 100 meters, 20 mm at 200 meters, etc., or similarly 0.39 inches at 100 m, 0.78 inches at 200 m, etc.
Subtensions in mrad based optics are particularly useful together with target sizes and shooting distances in
metric units. The most common scope adjustment increment in mrad based rifle scopes is 0.1 mrad, which are sometimes called "one centimeter clicks" since 0.1 mrad equals exactly 1 cm at 100 meters, 2 cm at 200 meters, etc. Similarly, an adjustment click on a scope with 0.2 mrad adjustment will move the point of bullet impact 2 cm at 100 m and 4 cm at 200 m, etc.
When using a scope with both mrad adjustment and a reticle with mrad markings (called a mrad/mrad scope), the shooter can spot his own bullet impact and easily correct the sight if needed. If the shot was a miss, the mrad reticle can simply be used as a "ruler" to count the number of milliradians the shot was off target. The number of milliradians to correct is then multiplied by ten if the scope has 0.1 mrad adjustments. If for instance the shot was 0.6 mrad to the right of the target, 6 clicks will be needed to adjust the sight. This way there is no need for math, conversions, knowledge of target size or distance. This is true for a first focal plane scope at all magnifications, but a variable second focal plane must be set to a given magnification (usually its maximum magnification) for any mrad scales to be correct.
When using a scope with mrad adjustments, but without mrad markings in the reticle (i.e. a standard duplex cross-hair on a hunting or benchrest scope), sight correction for a known target subtension and known range can be calculated by the following formula, which utilizes the fact that an adjustment of 1 mrad changes the impact as many millimeters as there are meters:
For instance:
* = 0.4 mrad, or 4 clicks with a mrad adjustment scope.
* = 0.05 mrad, or 1 click with a 0.05 mrad adjustment scope.
In firearm optics, where 0.1 mrad per click is the most common mrad based adjustment value, another common rule of thumb is that an adjustment of mrad changes the impact as many centimeters as there are hundreds of meters. In other words, 1 cm at 100 meters, 2.25 cm at 225 meters, 0.5 cm at 50 meters, etc. See the table below
Adjustment range and base tilt

The horizontal and vertical adjustment range of a firearm sight is often advertised by the manufacturer using mrads. For instance a rifle scope may be advertised as having a vertical adjustment range of 20 mrad, 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 mrad vertical and 10 mrad horizontal adjustment. Elevation differ between models, but about 10–11 mrad are common in hunting scopes, while scopes made for
long range shooting
Long range shooting is a collective term for shooting sport, shooting disciplines where the distance to the target is significant enough that the shooter has to put effort into calculating external ballistics, various ballistic factors, esp ...
usually have an adjustment range of 20–30 mrad (70–100 moa).
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 mrad 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:
:
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 mount
Scope mounts are rigid implements used to attach (typically) a telescopic sight or other types of optical sights onto a firearm. The mount can be made integral to the scope body (such as the Zeiss rail) or, more commonly, an external fitting t ...
s 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 mrad, which equals 3 m at 1000 m (or 0.3 m at 100 m)
* 6 mrad, which equals 6 m at 1000 m (or 0.6 m at 100 m)
* 9 mrad, 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:
:
The adjustment range needed to shoot at a certain distance varies with firearm, caliber and load. For example, with a certain
.308 load and firearm combination, the bullet may drop 13 mrad at 1000 meters (13 meters). To be able to reach out, one could either:
* Use a scope with 26 mrad of adjustment in a neutral mount, to get a usable adjustment of = 13 mrad
* Use a scope with 14 mrad of adjustment and a 6 mrad tilted mount to achieve a maximum adjustment of + 6 = 13 mrad
Shot groupings
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
millimeter
330px, Different lengths as in respect of the electromagnetic spectrum, measured by the metre and its derived scales. The microwave is between 1 metre to 1 millimetre.
The millimetre (American and British English spelling differences#-re, -er, i ...
s with a
caliper and dividing by the shooting distance in meters. This way, using milliradians, one can easily compare shot groupings or target difficulties at different shooting distances.
:
If the firearm is attached in a fixed mount and aimed at a target, the shot grouping measures the firearm's 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, especially when competitors are using the same match grade firearms and ammunition.
Range estimation with mrad reticles
Many telescopic sights used on rifles have
reticle
A reticle or reticule, also known as a graticule or crosshair, is a pattern of fine lines or markings built into the eyepiece of an optical device such as a telescopic sight, spotting scope, theodolite, optical microscope or the electronic v ...
s that are marked in mrad. This can either be accomplished with lines or dots, and the latter is generally called mil-dots. The mrad reticle serves two purposes, range estimation and trajectory correction.
With a mrad 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 milliradians 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 milliradians, can be used to adjust the aiming point. Generally mrad-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. 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.

Angle can be used for either calculating target size or range if one of them is 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 mrad wide, a fist 150 mrad and a spread hand 300 mrad.
Milliradian reticles often have dots or marks with a spacing of 1 mrad in between, but graduations can also be finer and coarser (i.e. 0.8 or 1.2 mrad).
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 mrad 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.
:
The same holds true for calculating target distance in kilometers using target size in meters.
:
Also, in general the same unit can be used for subtension and range if multiplied with a factor of thousand, i.e.
:
If using the
imperial units
The imperial system of units, imperial system or imperial units (also known as British Imperial or Exchequer Standards of 1826) is the system of units first defined in the British Weights and Measures Act 1824 and continued to be developed thr ...
yards for distance and inches for target size, one has to multiply by a factor of ≈ 27.78, since there are 36 inches in one yard.
:
If using the metric unit meters for distance and the imperial unit inches for target size, one has to multiply by a factor of 25.4, since one inch is defined as 25.4 millimeters.
:
Practical examples
Land Rovers are about 3 to 4 m long, "smaller tank" or
APC/
MICV at about 6 m (e.g.
T-34
The T-34 is a Soviet medium tank from World War II. When introduced, its 76.2 mm (3 in) tank gun was more powerful than many of its contemporaries, and its 60-degree sloped armour provided good protection against Anti-tank warfare, ...
or
BMP) and about 10 m for a "big tank." 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. 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 mrad and minutes of arc
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. It is preferred to either have both a mrad reticle and mrad adjustment (mrad/mrad), 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 mrad reticle and arcminute adjustment, one way to make use of the reticle for shot corrections is to exploit that 14′ approximately equals 4 mrad, and thereby multiplying an observed corrections in mrad by a fraction of when adjusting the turrets.
Conversion table for firearms

In the table below conversions from mrad to metric values are exact (e.g. 0.1 mrad equals exactly 1 cm at 100 meters), while conversions of minutes of arc to both metric and imperial values are approximate.
* 0.1 mrad equals exactly 1 cm at 100 m
* 1 mrad ≈ 3.44′, so mrad ≈ ′
* 1′ ≈ 0.291 mrad (or 2.91 cm at 100 m, approximately 3 cm at 100 m)
Definitions for maps and artillery

Because of the definition of pi, in a circle with a diameter of one there are 2000 milliradians () per full turn. In other words, one real milliradian covers just under of the circumference of a circle, which is the definition used by telescopic rifle sight manufacturers in reticles for
stadiametric rangefinding.
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 sometimes referred to as "angular mils", and are:
* of a circle in
NATO
The North Atlantic Treaty Organization (NATO ; , OTAN), also called the North Atlantic Alliance, is an intergovernmental organization, intergovernmental Transnationalism, transnational military alliance of 32 Member states of NATO, member s ...
countries.
* of a circle in
the former Soviet Union and
Finland
Finland, officially the Republic of Finland, is a Nordic country in Northern Europe. It borders Sweden to the northwest, Norway to the north, and Russia to the east, with the Gulf of Bothnia to the west and the Gulf of Finland to the south, ...
(Finland phasing out the standard in favour of the NATO standard).
* of a circle in
Sweden
Sweden, formally the Kingdom of Sweden, is a Nordic countries, Nordic country located on the Scandinavian Peninsula in Northern Europe. It borders Norway to the west and north, and Finland to the east. At , Sweden is the largest Nordic count ...
. The Swedish term for this is ''streck'', literally "line".
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
East Germany
East Germany, officially known as the German Democratic Republic (GDR), was a country in Central Europe from Foundation of East Germany, its formation on 7 October 1949 until German reunification, its reunification with West Germany (FRG) on ...
from 1969 to 1976 is calibrated for the eastern bloc 6000 mil circle.
Various symbols have been used to represent angular mils for compass use:
* mil, MIL and similar abbreviations are often used by militaries in the English speaking part of the world.
* ‰, called "artillery
per mille
The phrase per mille () indicates parts per thousand. The associated symbol is , similar to a per cent sign but with an extra zero in the division (mathematics), divisor.
Major dictionaries do not agree on the spelling, giving other options o ...
s" (German: ''Artilleriepromille''), a symbol used by the
Swiss Army.
* ¯, called "artillery line" (German: ''artilleristische Strich''), a symbol used by the
German Army
The German Army (, 'army') is the land component of the armed forces of Federal Republic of Germany, Germany. The present-day German Army was founded in 1955 as part of the newly formed West German together with the German Navy, ''Marine'' (G ...
(not to be confused with
Compass Point (German: ''Nautischer Strich'', 32 "nautical lines" per circle) which sometimes use the same symbol. However, the
DIN standard (DIN 1301 part 3) is to use ¯ for artillery lines, and " for nautical lines.)
*
₥, called "thousandths" (French: ''millièmes''), a symbol used on some older French compasses.
*
v (Finnish: ''piiru'', Swedish: ''delstreck''), a symbol used by the
Finnish Defence Forces
The Finnish Defence Forces (FDF) (; ) are the military of Finland. The Finnish Defence Forces consist of the Finnish Army, the Finnish Navy, and the Finnish Air Force. In wartime, the Finnish Border Guard becomes part of the Finnish Defence For ...
for the standard Warsaw Pact mil.
Sometimes just marked as v if superscript is not available.
Conversion table for compasses
File:Kompassrose.svg, A 360 degree and 6400 NATO mil compass rose.
File:RECTA full syst.jpg, Swiss Army compass with 6400 ‰ ("artillery per milles")
File:Liquid filled compass.jpg, US Army
The United States Army (USA) is the primary land service branch of the United States Department of Defense. It is designated as the Army of the United States in the United States Constitution.Article II, section 2, clause 1 of the United Stat ...
compass with scales both in 360 degrees and NATO 6400 mil
File:Boussole fantassin russe.jpg, Soviet Army wrist compass with two (opposite) scales, 360 degrees clockwise and 6000 Soviet mil counterclockwise.
Use in artillery sights
Artillery uses angular measurement in gun laying, the azimuth between the gun and its target many kilometers 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 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. A mil is a meter at a range of one thousand meters (for example, to move the impact of an artillery round 100 meters by a gun firing from 3 km away, it is necessary to shift the direction by 100/3 = 33.3 mils.)
Other scientific and technological uses
The milliradian (and
other SI multiples) is also used in other fields of
science
Science is a systematic discipline that builds and organises knowledge in the form of testable hypotheses and predictions about the universe. Modern science is typically divided into twoor threemajor branches: the natural sciences, which stu ...
and
technology
Technology is the application of Conceptual model, conceptual knowledge to achieve practical goals, especially in a reproducible way. The word ''technology'' can also mean the products resulting from such efforts, including both tangible too ...
for describing small angles, i.e. measuring alignment,
collimation, and
beam divergence in
optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of optical instruments, instruments that use or Photodetector, detect it. Optics usually describes t ...
, and
accelerometer
An accelerometer is a device that measures the proper acceleration of an object. Proper acceleration is the acceleration (the rate of change (mathematics), rate of change of velocity) of the object relative to an observer who is in free fall (tha ...
s and
gyroscope
A gyroscope (from Ancient Greek γῦρος ''gŷros'', "round" and σκοπέω ''skopéō'', "to look") is a device used for measuring or maintaining Orientation (geometry), orientation and angular velocity. It is a spinning wheel or disc in ...
s in
inertial navigation system
An inertial navigation system (INS; also inertial guidance system, inertial instrument) is a navigation device that uses motion sensors (accelerometers), rotation sensors (gyroscopes) and a computer to continuously calculate by dead reckoning th ...
s.
See also
*
Scandinavian mile, a unit of length common in
Norway
Norway, officially the Kingdom of Norway, is a Nordic countries, Nordic country located on the Scandinavian Peninsula in Northern Europe. The remote Arctic island of Jan Mayen and the archipelago of Svalbard also form part of the Kingdom of ...
and
Sweden
Sweden, formally the Kingdom of Sweden, is a Nordic countries, Nordic country located on the Scandinavian Peninsula in Northern Europe. It borders Norway to the west and north, and Finland to the east. At , Sweden is the largest Nordic count ...
, but not
Denmark
Denmark is a Nordic countries, Nordic country in Northern Europe. It is the metropole and most populous constituent of the Kingdom of Denmark,, . also known as the Danish Realm, a constitutionally unitary state that includes the Autonomous a ...
, today standardised as 10
kilometer
The kilometre ( SI symbol: km; or ), spelt kilometer in American and Philippine English, is a unit of length in the International System of Units (SI), equal to one thousand metres (kilo- being the SI prefix for ). It is the preferred mea ...
s.
*
Thousandth of an inch
A thousandth of an inch is a derived unit of length in a system of units using inches. Equal to of an inch, a thousandth is commonly called a thou (used for both singular and plural) or, particularly in North America, a mil (plural mils).
Th ...
, an inch-based unit often called a ''thou'' or a ''mil''.
*
Circular mil, a unit of area, equal to the area of a circle with a diameter of one thousandth of an inch.
*
Square mil, a unit of area, equal to the area of a square with sides of length of one thousandth of an inch.
Footnotes
References
External links
*
*
* {{cite web , url=http://compassmuseum.com/diverstext/divisions.htm , title=Compassipedia , website=The Online Compass Museum
Units of plane angle
Decimalisation
Optical devices