Angular mil
Encyclopedia
An angular mil, also mil, is a unit
of angle
. All versions of the angular mil are approximately the same size as a trigonometric milliradian.
' (circle/400) under various names having considerable popularity in much of northern Europe, the grad was based on dividing a right angle into 100 units. However, Imperial Russia used a different approach, dividing a circle into equilateral triangles 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 (circle/6400) for use with artillery sights instead of decigrades (circle/4000). The United Kingdom was also trialling 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 (circle/6400). After the Bolshevik Revolution and the adoption of the metric system of measurement (e.g. artillery replaced 'units of base' with metres) the Red Army expanded the 600 unit circle into a 6000 mil one, hence the Russian mil has nothing to do with milliradians as its origin.
In the 1950s, NATO adopted metric units of measurement for land and general use. Mils, metres and kilograms became standard, although degrees remained in use for naval and air purposes, reflecting civil practices.
This allows a user to dispense with trigonometry
and use simple ratios to determine size and distance with acceptable accuracy for rifle and short distance artillery calculations.
Since a radian
is mathematically defined as the angle formed when the length of a circular arc equals the radius of the circle, a trigonometric milliradian (mrad), is the angle formed when the length of a circular arc equals 1/1000 of the radius of the circle. Since the radian expresses a ratio, it is independent of the units of length used.
for skinny triangles
shows that the angle in radians approximates to the sine
of the angle. More usefully an angular mil or mil subtends about 1 m at a range of 1,000 m.
Angle can be used for both calculating size or range. 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 by using calibrated optics or quite approximately ones fingers & hands. With an outstretched arm one finger is approximately 30 mils wide, a fist 150 mils and a spread hand 300 mils.
Angle can be used for calculating range. For objects of known size the range is the size divided by the angle. Land Rovers
are about 3 to 4 m long, "smaller tank" or APC
/MICV
at ~6 m (e.g. T-34
or BMP
) and ~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 ~100 m. A modern tank would have to be at a bit over 300 m.
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.
Note: Do not confuse the angular mil with the MOA (minute of angle
). 1 mil = 3.438 MOA
s), together with their manual plotting devices, firing tables and fire control computers.
s used on rifles have reticles that are marked in angular mils, and these are generally called Mil-dot scopes. The Mil-dots serve two purposes, range estimation and trajectory correction.
With a Mil-dot 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-dot 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-dot reticle-equipped scopes are most suited for long shots under uncertain conditions, such as those encountered by military and law enforcement sniper
s, 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.
, (which includes most of the world's military forces by whom this system was developed), can much more easily use a Mil-dot reticle since the mental arithmetic
is much simpler. It is always decimal.
To determine the distance or range to a target of known size at an unknown distance this formula can be applied:
where:
This is otherwise simplified as:
where:
When used with some riflescopes of variable objective magnification and fixed reticle magnification (where the reticle is in the second focal plane), this can be modified to:
where:
(However, a user should verify this with their individual scope since some are not calibrated at 10x.)
To determine the size of a target at a known distance: (Object size in inches) = 36 / 1000 x (Distance in yards) x Mils
Knowing these relationships, you can set up a target range by drawing a vertical or horizontal line onto our target that is 3.6 inches long then backing up until the line is exactly one mil long in your scope for a 100 yard range, 7.2 inches for 200 yards, 10.8 inches for 300 yards, etc.
Known Target Size (8 inch) Ranging:
of a circle in NATO countries. The “real” trigonometric unit of angular measurement of a circle in use by telescopic sight manufacturers using (stadiametric) rangefinding in reticles. of a circle in the former Soviet Union
and Finland
(Finland phasing out the standard in favour of the NATO standard). of a circle in Sweden
. The Swedish term for this is streck, literally "line". Sweden has not been part of NATO nor the Warsaw Pact
. Note however that Sweden is changing its map grid systems and angular measurement to those used by NATO, so the "streck" measurement is about to become obsolete.
Units of measurement
A unit of measurement is a definite magnitude of a physical quantity, defined and adopted by convention and/or by law, that is used as a standard for measurement of the same physical quantity. Any other value of the physical quantity can be expressed as a simple multiple of the unit of...
of angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
. All versions of the angular mil are approximately the same size as a trigonometric milliradian.
History
The milliradian was first identified in the mid nineteenth Century. Degrees and minutes were the usual units of angular measurement but others were being proposed, with 'gradsGrad (angle)
The gradian is a unit of plane angle, equivalent to of a turn. It is also known as gon, grad, or grade . One grad equals of a degree or of a radian...
' (circle/400) under various names having considerable popularity in much of northern Europe, the grad was based on dividing a right angle into 100 units. However, Imperial Russia used a different approach, dividing a circle into equilateral triangles and hence 600 units to a circle.
Around the time of the start of World War I
World War I
World War I , which was predominantly called the World War or the Great War from its occurrence until 1939, and the First World War or World War I thereafter, was a major war centred in Europe that began on 28 July 1914 and lasted until 11 November 1918...
, France was experimenting with the use of milliemes (circle/6400) for use with artillery sights instead of decigrades (circle/4000). The United Kingdom was also trialling 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 (circle/6400). After the Bolshevik Revolution and the adoption of the metric system of measurement (e.g. artillery replaced 'units of base' with metres) the Red Army expanded the 600 unit circle into a 6000 mil one, hence the Russian mil has nothing to do with milliradians as its origin.
In the 1950s, NATO adopted metric units of measurement for land and general use. Mils, metres and kilograms became standard, although degrees remained in use for naval and air purposes, reflecting civil practices.
Mathematical principle
Use of the the Mil-dot system is possible because it is concerned with small angles, and at small angles:This allows a user to dispense with trigonometry
Trigonometry
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...
and use simple ratios to determine size and distance with acceptable accuracy for rifle and short distance artillery calculations.
Since a radian
Radian
Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...
is mathematically defined as the angle formed when the length of a circular arc equals the radius of the circle, a trigonometric milliradian (mrad), is the angle formed when the length of a circular arc equals 1/1000 of the radius of the circle. Since the radian expresses a ratio, it is independent of the units of length used.
Use
The angular mil is commonly used by military organizations. Its relationship to the trigonometric radian gives rise to the handy property of subtension - that "One mil approximately subtends one metre at a distance of one thousand metres". More formally the small angle approximationSmall angle approximation
The small-angle approximation is a useful simplification of the basic trigonometric functions which is approximately true in the limit where the angle approaches zero. They are truncations of the Taylor series for the basic trigonometric functions to a second-order approximation...
for skinny triangles
Skinny triangle
A skinny triangle in trigonometry is a triangle whose height is much greater than its base. The solution of such triangles can be greatly simplified by using the approximation that the sine of a small angle is equal to the angle in radians...
shows that the angle in radians approximates to the sine
Sine
In mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....
of the angle. More usefully an angular mil or mil subtends about 1 m at a range of 1,000 m.
Angle can be used for both calculating size or range. 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 by using calibrated optics or quite approximately ones fingers & hands. With an outstretched arm one finger is approximately 30 mils wide, a fist 150 mils and a spread hand 300 mils.
Angle can be used for calculating range. For objects of known size the range is the size divided by the angle. Land Rovers
Land Rover Series
The Land Rover Series I, II, and III are off-road vehicles produced by the British manufacturer Land Rover that were inspired by the US-built Willys Jeep...
are about 3 to 4 m long, "smaller tank" or APC
Armoured personnel carrier
An armoured personnel carrier is an armoured fighting vehicle designed to transport infantry to the battlefield.APCs are usually armed with only a machine gun although variants carry recoilless rifles, anti-tank guided missiles , or mortars...
/MICV
Infantry fighting vehicle
An infantry fighting vehicle , also known as a mechanized infantry combat vehicle , is a type of armoured fighting vehicle used to carry infantry into battle and provide fire support for them...
at ~6 m (e.g. T-34
T-34
The T-34 was a Soviet medium tank produced from 1940 to 1958. Although its armour and armament were surpassed by later tanks of the era, it has been often credited as the most effective, efficient and influential design of World War II...
or BMP
BMP-3
The BMP-3 is a Russian amphibious infantry fighting vehicle, successor to the BMP-1 and BMP-2. BMP stands for Boyevaya Mashina Pekhoty .- Production history :...
) and ~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 ~100 m. A modern tank would have to be at a bit over 300 m.
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.
Note: Do not confuse the angular mil with the MOA (minute of angle
Minute of Angle
Minute of angle is the measurement of a ballistic round's deviation from its initial heading due to gravity and/or the effect of air resistance on velocity. Informally known as a "Bullet's Trajectory" or "the rainbow effect". Long range weapons must account for this effect because a fired round...
). 1 mil = 3.438 MOA
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 quadrantQuadrant (instrument)
A quadrant is an instrument that is used to measure angles up to 90°. It was originally proposed by Ptolemy as a better kind of astrolabe. Several different variations of the instrument were later produced by medieval Muslim astronomers.-Types of quadrants:...
s), together with their manual plotting devices, firing tables and fire control computers.
Telescopic sights
Many telescopic sightTelescopic sight
A telescopic sight, commonly called a scope, is a sighting device that is based on an optical refracting telescope. They are equipped with some form of graphic image pattern mounted in an optically appropriate position in their optical system to give an accurate aiming point...
s used on rifles have reticles that are marked in angular mils, and these are generally called Mil-dot scopes. The Mil-dots serve two purposes, range estimation and trajectory correction.
With a Mil-dot 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
External ballistics
External ballistics is the part of the science of ballistics that deals with the behaviour of a non-powered projectile in flight. External ballistics is frequently associated with firearms, and deals with the behaviour of the bullet after it exits the barrel and before it hits the target.-Forces...
), converted back into angular mils, can be used to adjust the aiming point. Generally Mil-dot 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-dot reticle-equipped scopes are most suited for long shots under uncertain conditions, such as those encountered by military and law enforcement sniper
Sniper
A sniper is a marksman who shoots targets from concealed positions or distances exceeding the capabilities of regular personnel. Snipers typically have specialized training and distinct high-precision rifles....
s, varmint hunters
Varmint rifle
Varmint rifle is an American English term for a small-caliber firearm or high-powered air gun primarily used for varmint hunting — killing non-native or non-game animals such as rats, house sparrows, starling, crows, ground squirrels, gophers, jackrabbits, marmots, groundhogs, porcupine,...
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.
Metric Mil-dot formula
Users of the metric systemMetric system
The metric system is an international decimalised system of measurement. France was first to adopt a metric system, in 1799, and a metric system is now the official system of measurement, used in almost every country in the world...
, (which includes most of the world's military forces by whom this system was developed), can much more easily use a Mil-dot reticle since the mental arithmetic
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
is much simpler. It is always decimal.
To determine the distance or range to a target of known size at an unknown distance this formula can be applied:
where:
- D = distance or range to the target in meters
- S = size of the target in centimeters (known width or height of the target)
- mil = number of Mil-dots
This is otherwise simplified as:
where:
- D = distance to the target in meters
- S = actual size of target in millimeters
- mil = scope size of target measured in Mil-dots
When used with some riflescopes of variable objective magnification and fixed reticle magnification (where the reticle is in the second focal plane), this can be modified to:
where:
- D = distance to the target in meters
- S = actual size of target in millimeters
- mil = scope size of target measured in Mil-dots
- mag = scope magnification
(However, a user should verify this with their individual scope since some are not calibrated at 10x.)
Imperial and US customary units Mil-dot formula
To determine the distance to a target of known size: (Distance in yards) = 1000 / 36 x (Object size in inches) / MilsTo determine the size of a target at a known distance: (Object size in inches) = 36 / 1000 x (Distance in yards) x Mils
Knowing these relationships, you can set up a target range by drawing a vertical or horizontal line onto our target that is 3.6 inches long then backing up until the line is exactly one mil long in your scope for a 100 yard range, 7.2 inches for 200 yards, 10.8 inches for 300 yards, etc.
Known Target Size (8 inch) Ranging:
| Target, inches | Mils | Distance, yards |
|---|---|---|
| 8 | 0.1 | 2222 |
| 8 | 0.2 | 1111 |
| 8 | 0.3 | 741 |
| 8 | 0.4 | 556 |
| 8 | 0.5 | 444 |
| 8 | 0.6 | 370 |
| 8 | 0.7 | 317 |
| 8 | 0.8 | 278 |
| 8 | 0.9 | 247 |
| 8 | 1.0 | 222 |
| 8 | 1.1 | 202 |
| 8 | 1.2 | 185 |
| 8 | 1.3 | 171 |
| 8 | 1.4 | 159 |
| 8 | 1.5 | 148 |
| 8 | 2.0 | 111 |
| 8 | 2.2 | 101 |
| 8 | 3.0 | 74 |
| 8 | 4.0 | 56 |
| 8 | 5.0 | 44 |
| 8 | 6.0 | 37 |
| 8 | 7.0 | 32 |
| 8 | 8.0 | 28 |
| 8 | 9.0 | 25 |
| 8 | 10.0 | 22 |
Definitions of the angular mil
There are 2000π milliradians (≈ 6283.185 mrad) in a circle. So a milliradian is just under 1⁄6283 of a circle, or ≈ 3.438 minutes of arc. Each of the definitions of the angular mil are similar to that value but are easier to divide into many parts.of a circle in NATO countries. The “real” trigonometric unit of angular measurement of a circle in use by telescopic sight manufacturers using (stadiametric) rangefinding in reticles. of a circle in the former Soviet Union
Soviet Union
The Soviet Union , officially the Union of Soviet Socialist Republics , was a constitutionally socialist state that existed in Eurasia between 1922 and 1991....
and Finland
Finland
Finland , officially the Republic of Finland, is a Nordic country situated in the Fennoscandian region of Northern Europe. It is bordered by Sweden in the west, Norway in the north and Russia in the east, while Estonia lies to its south across the Gulf of Finland.Around 5.4 million people reside...
(Finland phasing out the standard in favour of the NATO standard). of a circle in Sweden
Sweden
Sweden , officially the Kingdom of Sweden , is a Nordic country on the Scandinavian Peninsula in Northern Europe. Sweden borders with Norway and Finland and is connected to Denmark by a bridge-tunnel across the Öresund....
. The Swedish term for this is streck, literally "line". Sweden has not been part of NATO nor the Warsaw Pact
Warsaw Pact
The Warsaw Treaty Organization of Friendship, Cooperation, and Mutual Assistance , or more commonly referred to as the Warsaw Pact, was a mutual defense treaty subscribed to by eight communist states in Eastern Europe...
. Note however that Sweden is changing its map grid systems and angular measurement to those used by NATO, so the "streck" measurement is about to become obsolete.