MIL Vs MRAD Vs NATO-MIL [mil.wndsn.com]

MIL vs. MRAD vs. NATO-MIL

There are various systems for working with MIL and various definitions leading to different values for 1 MIL, depending on where it is used, and in which context.

In mathematics and all natural sciences, the angular distance (angular separation, apparent distance, or apparent separation) between two points, as observed from a location different from either of these points, is the size of the angle between the two directions originating from the observer and pointing towards these two points.

This angular distance, or size, is measured in degrees or units specified as fractions of a degree, like the MIL.

The scientific MIL, also called MRAD, is a 1000th of a radian (which leads to an odd value for a full circle), while the military MIL is based on dividing the circle into an even number that splits nicely into decimal fractions.

Clarification of terms

MIL usually denotes MRAD, MILliradians; thousandths of a radian. Militaries around the world have historically used rounded versions of the MRAD, the NATO-MIL and the German army's 'Artilleristischer Strich' divide the full circle (or turn) by 6400 -- instead of the MRAD's more accurate but also harder to subdivide 6283.x -- while the Warsaw Pact established 6000 MILs in a turn and the Swedish divide it into 6300 "streck".

Definitions

1 MRAD is defined as 1000/rad (0.001 radian) (an SI-derived unit) hence there are 1000 MILs per radian, hence 2000π or approximately 6283.185 milliradians in 360° or approximately 0.057296° for 1 MRAD.
1 NATO-MIL is defined as 360°/6400 or exactly 0.05625°.

Conversions^[1]

1 MRAD = 1.02 NATO-MIL ≅ 0.057296°
1 NATO-MIL = 0.98 MRAD = 0.05625°
1° = 17.453 MRAD ≅ 17.778 NATO-MIL

0.05625 / 0.057296 = 0.98174392627    
0.057296 / 0.05625 = 1.01859555556    
Difference = 1.8595%

Summary

A full circle is made up of 360°.
A full circle is 2π radians, hence each radian measuring 57.3°.
Each radian (180/π ≅ 57.3°) is made up of 1000 milliradians (MRAD).
A full circle is approximately 6283 MRAD.
To simplify, NATO rounds these to 6400 NATO-MIL in a circle.

Why MIL?

(Excerpt from: Wndsn Quadrant Telemeters: Graphical Telemetry Computers. Low Tech Distance & Altitude Nomographs. Instruments for the Mastery of Time and Space)

Use of the milliradian is practical because when using radians, the small angle approximation shows that the angle approximates to the sine of the angle, that is sinθ ≃ θ. This allows us to simplify trigonometry and use mere ratios to determine size and distance with high accuracy for rifle and short distance artillery calculations by using the property of subtension:

One MIL approximately subtends one meter at a distance of one thousand meters.^[2]

A good approximation is using the definition of a radian and the simplified formula for milliradian subtension:

θrad = subtension / 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. 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:

d = s × 1000 / MIL

where: d is the distance to the object; s is the size of the object observed; and MIL is the apparent size of the object observed. Note that this formula is actually unitless -- the same formula takes yard or meters or any other unit, provided that the object and distance are input and read in the same unit.

In Germany, the formula is known as MKS-Formel

M = K × S or K = M / S or S = M / K

where: M is the size of the object observed (in meters); K is the distance to the target (in km); and S is the angular size of the object observed (in MIL).

distance in km = target in meters / angle in MIL

with the multiplier 1000 being "built-in" by using km and meters, respectively.

Multipliers

Another interesting property of MIL (MRAD) is that some values provide simple multiplication opportunities; knowing the size of the target, it becomes possible to directly multiply with the MIL-value to acquire the distance for certain measurements:

 10 MIL = x100 = 0.5729°
 20 MIL = x 50
 40 MIL = x 25
 50 MIL = x 20
100 MIL = x 10 = 5.729°

Wndsn Usage

The civilian Telemeter model uses degrees divided into tenths for maximum resolution. Since a MIL (at ~0.05°) is smaller than this resolution, degrees are sufficient. The military Telemeter model with a scale of 0 to 150 MIL is available in a precision of 1 MIL for those who prefer to calculate in MIL (NATO) or process values in MIL from sighting reticles or similar.

For actual nomogram operation on the Telemeter, it does not matter whether the degree scale or the MIL scale (on the Telemeters where they are present) is used. The relationship between degrees and MIL is linear and conversion from one unit to another is trivial (see above).

For rangefinding calculations of values ranging from 0 to 6 MIL and for conversions between MIL (in this case MRAD) and MOA, we provide a graphical calculator that allows seamless switching between metric and imperial systems.

Calculators, and the instruments:

References:

Note that the difference of MRAD vs. NATO-MIL is, with 1.86%, smaller than the typical uncertainty of measurement and below the accuracy of most measuring techniques. Still, be aware of which unit is used, and make sure that it doesn't add to the overall measurement uncertainty. ↩
One MIL is equal to one 1000th of the target range, laid out on a circle that has the observer in its center and the target range as its radius. The number of MILs on a full such circle therefore always equals 2 × π × 1000, independent of target range. This means that an object which measures 1 MIL on the reticle is at a range that is in meters equal to the object's size in millimeters (e.g. an object of 100 mm at 1 MIL is 100 meters away). ↩