MIL vs MOA - The White Paper
by Gary Conway
MILS and MOA
First and foremost, a MIL is not the same as MOA. MILS are based on radian measure and MOA is based on degrees. Radians and degrees are simply two different ways of measuring angles. Additionally MIL does not stand for military, instead it means milli-radians. MILS and MOA are simply two distinct and different ways of measuring angles and subsequently, range. They are based on two different, yet equivalent methods of measuring angles, radians and degrees.

MIL = 1 milliradian, or 1/1000 radian
MOA = Minute Of Arc or arcminute = 1/60°

CONVERSION FACTORS

There are 2 x Π (approximately 6.28318531) radians in a circle

Where Π (pronounced PIE) is approximately 3.14159

Radians


2 x Π radians = 360°
1 radian = 360°/(2 x Π) =180°/ Π = 57.30°

Degrees


360° = 2 x Π radians
1° = (2 x Π) / 360 radians = Π / 180 radians = 0.0175 radians

Radians are a very handy and natural method of measuring angles by virtue of the distinct relationship between radius and angle. An angle measuring 1 radian subtends an arc equal in length to the radius of the circle as shown in the image below. This is the very fact that makes MIL measurements ideal for ranging; that there is a direct correspondence between the MIL height of an object and its range.

The term radian first appeared in print on June 5, 1873 , in examination questions by James Thomson, brother of Lord Kelvin . He used the term as early as 1871, while in 1869, Thomas Muir hesitated between rad, radial and radian. In 1874 , Muir adopted radian after a consultation with James Thomson. The concept of a radian measure, as opposed to the degree of an angle, should probably be credited to Roger Cotes in 1714. He had the radian in everything but name, and he recognized its naturalness as a unit of angular measure.

Minute of Arc (MOA)
A minute of arc, minute of angle, arcminute, or MOA is a unit of angular measurement , equal to one sixtieth (1/60) of one degree. Its subdivision arcsecond comprises one sixtieth of an arcminute. Since one degree is defined as one three hundred and sixtieth (1/360) of a circle, 1 arcminute is 1/21600 of the amount of arc in a closed circle, or (Π/10800) radians.

' = minutes of angle (60 minutes/degree) (21600 minutes/circle)
" = seconds of angle (60 seconds/minute) (3600 seconds/degree) (1296000 seconds/circle)

The term "minute-of-arc" (MOA) is used regularly by target shooters at the range, although it is likely that relatively few have a full grasp of its meaning. One MOA is approximately 1 inch at 100 yards, which turns out to be a very handy and useful relationship. If you have a 1/4 MOA sight, then each click will move the bullet strike 1/4 inch at 100 yards. Likewise, 1 MOA is 2 inches at 200 yards, 3 inches at 300 yards and so on. Ok, fine, but how do we know this? The answer will involve some math, so plant your butt, grab a beer and relax whilst we trudge through the lamery.

To calculate the range to a target, we can use our knowledge of the relationship between the subtended angle and the range. Trigonometry gives us the mathematical relation we need, namely the tangent function, which is defined by parts of a triangle. The tangent is defined as length of the side opposite the angle divided by the length of the side adjacent to the angle, eg. tan(angle) = opposite / adjacent. In our diagram below, the tangent is defined as HEIGHT / RANGE.

The caveat here is that the units for both height and range must be consistent, that is, both must be inches, feet, yards, lightyears etc.

Now let's apply what we know to proving that 1 MOA is indeed 1 inch at 100 yards.

RANGE (to target) = 100 yards
HEIGHT (of target) = 1 inch or 1/36 yard

therefore our equation is..

tan(angle) = HEIGHT / RANGE = (1/36) / 100 = .000277778

OK ,great, but so what? Well, the number we just calculated is the TANGENT of the angle, but what we want is the actual angle itself. To do that we must calculate the ARC-TANGENT, this is the mathematical function that gives us the angle when we know the tangent of the angle. ARCTAN(.00027778) = .015915494 degrees, we can convert this to MOA by multiplying by 60, since there are 60 minutes per degree. MOA ("angle" in the equation), then is .954929634, pretty close to 1 MOA huh.

Let's look at it from another angle (no pun intended, well, sorta)...

We can rewrite our equation as follows...

tan(angle) x RANGE = HEIGHT

our angle is 1 MOA, or 1/60 degree, so..

tan(1/60)=.000290888, so...
tan(1/60) x 100 yards = .000277778 x 100 = 1.047197581 inches (pretty close to 1 inch huh, less than 1/16 inch off)

Most sighting systems utilize adjustments that are calibrated in MOA. Each click of the adjustment knob represents 1/4 MOA or 1/8 MOA on most systems. Shooters typically refer to these adjustments as to how far they move the bullet strike at 100 yards. Since we now know that 1 MOA = 1 inch at 100 yards, we can deduce the following...

1/4 MOA sights = 1/4 inch at 100 yards for each click
1/8 MOA sights = 1/8 inch at 100 yards for each click

MILS
The "MIL" in "Mil-Dot" does not stand for "Military"; it stands for "milliradian." The radian is a unitless measure which is equivalent, in use, to degrees. It tells us how far around a circle we have gone. 2 PI radians = 360 degrees. Using 3.14 as the value of PI, 6.28 radians takes you all the way around a circle. We can use a cartesian coordinate system, whereby we use "x"- and "y"-values to define any point on the plane. Radians are used in a coordinate system called "polar coordinates." A point on the plane is defined, in the polar coordinate system, using the radian and the radius. The radian defines the amount of rotation and the radius gives the distance from the origin (in a negative or positive direction).

The radian is another measurement of rotation (the degree/minute/second-system being the first). This is the system used in the mil-dot reticle. We use the same equation that we used before. One milliradian = 1/1000 (.001) radians. With your calculator in "radians" mode, type .001 and hit the "tangent" button. Then multiply this by "distance to the target." Finally, multiply this by 36 to get inches subtended at the given distance.

Using our equation from above..

tangent(angle) x RANGE = HEIGHT

angle = 1 MIL = 1 milliradian = .001 radians

NOTE: we are using inches below, so we multiply RANGE (yards) by 36 to convert to inches

tangent(.001) x 100 x 36 = 3.6000012 inches

So, one milliradian is just over 3.6 inches at 100 yards or 7.2 inches at 200 yards.

MIL-DOT RETICLE

Mil-Dot Reticle

The mil-dot reticle was designed around the measurement unit of the milliradian. The dots, themselves, were designed with this in mind and the spacing of the dots was also based upon the milliradian. This allows the shooter to calculate the distance to an object of known height or width. Height of the target in yards divided by the height of the target in milliradians multiplied by 1000 equals the distance to the target in yards. For example, take a 6-foot-tall man (2 yards). Let's say that the top of his head lines up with one dot and his feet line up four dots down. So: (2/4)*1000 = 500 yards away. This same technique can be used to estimate lead on a moving target or to compensate for deflection on a windy day.

The distance from the center of one dot to the center of the next dot is 1 milliradian. The length of a dot is 1/4 milliradian or 3/4 MOA (Given this much information, one can determine that the distance between dots is 3/4 milliradian.). I use the term "length" because the mil-dot is not round. It is oblong. The "dots" on the vertical crosshair run oblong in the vertical direction. The dots on the horizontal crosshair run oblong in the horizontal direction. The width of each dot is an arbitrary distance and is not used for any practical purpose. Like a duplex reticle, the mil-dot reticle is thicker towards the edges and uses thin lines in the middle where the dots are located and the crosshairs cross. The distance between the opposite thick portions is 10 milliradians.

Some Useful Things To Note
1 MIL = 1 yard @ 1000 yards
1 MIL = 1 meter @ 1000 meters

For 100 yards...

1 MIL = 3.438 MOA
1 MIL = 3.6 inches
1 MOA = 1.0472 inches
.27778 MIL = 1 inch

Why Use MOA or MIL
Knowing a little geometry, or at least how to use it in simplified form, becomes rather valuable when talking about making range measurements. As this page outlines, knowing a little about angles and how they relate to target height and distance is an invaluable tool when talking about accuracy. Knowing that our sight systems are either calculated in MILS or MOA and how much each click on an adjustment means, coupled with our knowledge as just stated, we then are able to adjust our sight systems to zero in on any target within the usable range of our weapon.

Of course, we haven't EVEN talked about nature... wind, barometric pressure, air density etc. That's a whole other world series event.



[an error occurred while processing this directive]