Reference Materials
Tables, conversions, and the manual.
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
On Wndsn Telemeters degrees are used, graded into tenths for the maximum resolution. Since a MIL (at ~0.05°) is smaller than that resolution, degrees are sufficient. We do offer a special Telemeter model graded from 0 to 130 MIL in 2 MIL increments for those who prefer calculating in MIL (NATO) or have values obtained in MIL from sighting reticles, etc.
For actual nomogram operation on the Telemeter, it doesn't make a difference whether the degree scale or MIL scale (on those Telemeters that have them) is used, the relationship between degree and MIL is linear and converting from one to the other is trivial (see above).
For rangefinding computations of values in the 0-6 MIL range and conversions between MIL (MRAD in this case) and MOA, we provide a graphical calculator that offers seamless operation and switching between metric and imperial systems.
Calculators, and the instruments:
- Wndsn Quadrant Telemeter Tutorials
- Table: Radians in Degrees in MIL in MOA
- Table: Multipliers for a Given Angle
- Wndsn Navigation Tools
- Wndsn Web Calculators: Companion Tools to Wndsn Quadrant Telemeters
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). ↩