Milliradian

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 3.3 parts per billion) error for an angle of 0.1 mrad, for instance by assuming 0.1 mrad equals 1 cm at 100 m^[3]
• 0.03% error for 30 mrad, i.e. assuming 30 mrad equals 30 m at 1 km^[4]
• 2.9% error for 300 mrad, i.e. assuming 300 mrad equals 300 m at 1 km^[5]

More information ...

Calculation of percentage error for mrad
${\displaystyle \delta =100\%\times \left|{\frac {\theta _{\text{trig}}-\theta _{\text{rad}}}{\theta _{\text{trig}}}}\right|}$

Close

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:

• 4.72% error by assuming that an angle of 1′ equals 1 inch at 100 yd^[6]
• 4.75% error for 100′, i.e. assuming 100′ equals 100 in at 100 yd^[7]
• 7.36% error for 1000′, i.e. assuming 1000′ equals 1000 inches at 100 yd^[8]

More information , ...

Calculation of percentage error for arcminutes
${\displaystyle \delta =100\%\times \left|{\frac {{\text{arcminutes}}_{\text{ trig}}-{\text{inches per hundred of yards}}}{{\text{arcminutes}}_{\text{trig}}}}\right|}$ where ${\displaystyle {\text{arcminutes}}_{\text{ trig}}=\arctan {\frac {\text{inches per hundred of yards}}{3600}}\times {\frac {180}{\pi }}\times 60}$

Close

Common click values

General purpose scopes

Gradations (clicks) of ⁠1/4⁠′, ⁠1/10⁠ mrad and ⁠1/2⁠′ are used in general purpose sights for hunting, target and long range shooting 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

⁠0.25/10⁠ mrad, ⁠1/8⁠′ and ⁠0.5/10⁠ 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 ⁠0.5/10⁠ mrad or ⁠0.25/10⁠ 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

⁠1.5/10⁠ mrad and ⁠2/10⁠ 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 or angular diameter) 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:

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

For instance:

• ⁠20 mm/50 m⁠ = 0.4 mrad, or 4 clicks with a ⁠1/10⁠ mrad adjustment scope.
• ⁠50 mm/1000 m⁠ = 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 ⁠1/10⁠ 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 usually have an adjustment range of 20–30 mrad (70–100 moa).^{[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 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:

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

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 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:

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

The adjustment range needed to shoot at a certain distance 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 ⁠26 mrad/2⁠ = 13 mrad
• Use a scope with 14 mrad of adjustment and a 6 mrad tilted mount to achieve a maximum adjustment of ⁠14 mrad/2⁠ + 6 = 13 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.

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

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

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

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

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

If using the imperial units yards for distance and inches for target size, one has to multiply by a factor of 1000⁄36 ≈ 27.78, since there are 36 inches in one yard.

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

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.

${\displaystyle {\text{distance in meters}}={\frac {\text{target in inches}}{\text{angle in mrad}}}\times 25.4}$

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 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:

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

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:

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

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.

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 ⁠1/10⁠ mrad ≈ ⁠1/3⁠′
• 1′ ≈ 0.291 mrad (or 2.91 cm at 100 m, approximately 3 cm at 100 m)

Conversion table for compasses

• 
  A 360 degree and 6400 NATO mil compass rose.
• 
  Swiss Army compass with 6400 ‰ ("artillery per milles")
• 
  US Army compass with scales both in 360 degrees and NATO 6400 mil
• 
  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.)