Originally posted by Speedy Parker
Now how does that relate to ballistics and target acquisition?
The arcminute is commonly found in the firearms industry and literature, particularly concerning the precision of rifles, though the industry refers to it as minute of angle (MOA). It is especially popular as a unit of measurement with shooters familiar with the imperial measurement system because 1 MOA subtends a circle with a diameter of 1.047 inches (which is often rounded to just 1 inch) at 100 yards (2.66 cm at 91 m or 2.908 cm at 100 m), a traditional distance on American target ranges. The subtension is linear with the distance, for example, at 500 yards, 1 MOA subtends 5.235 inches, and at 1000 yards 1 MOA subtends 10.47 inches. Since many modern telescopic sights are adjustable in half (1/2), quarter (1/4) or eighth (1/8) MOA increments, also known as clicks, zeroing and adjustments are made by counting 2, 4 and 8 clicks per MOA respectively.
For example, if the point of impact is 3 inches high and 1.5 inches left of the point of aim at 100 yards (which for instance could be measured by using a spotting scope with a calibrated reticle), the scope needs to be adjusted 3 MOA down, and 1.5 MOA right. Such adjustments are trivial when the scope's adjustment dials have a MOA scale printed on them, and even figuring the right number of clicks is relatively easy on scopes that click in fractions of MOA. This makes zeroing and adjustments much easier:
To adjust a 1⁄2 MOA scope 3 MOA down and 1.5 MOA right, the scope needs to be adjusted 3 × 2 = 6 clicks down and 1.5 x 2 = 3 clicks right
To adjust a 1⁄4 MOA scope 3 MOA down and 1.5 MOA right, the scope needs to be adjusted 3 x 4 = 12 clicks down and 1.5 × 4 = 6 clicks right
To adjust a 1⁄8 MOA scope 3 MOA down and 1.5 MOA right, the scope needs to be adjusted 3 x 8 = 24 clicks down and 1.5 × 8 = 12 clicks right
Another common system of measurement in firearm scopes is the milliradian (mrad). Zeroing an mrad based scope is easy for users familiar with base ten systems. The most common adjustment value in mrad based scopes is 1/10 mrad (which approximates 1⁄3 MOA).
To adjust a 1/10 mrad scope 0.9 mrad down and 0.4 mrad right, the scope needs to be adjusted 9 clicks down and 4 clicks right (which equals approximately 3 and 1.5 MOA respectively). One thing to be aware of is that some MOA scopes, including some higher-end models, are calibrated such that an adjustment of 1 MOA on the scope knobs corresponds to exactly 1 inch of impact adjustment on a target at 100 yards, rather than the mathematically correct 1.047 inches. This is commonly known as the Shooter's MOA (SMOA) or Inches Per Hundred Yards (IPHY). While the difference between one true MOA and one SMOA is less than half of an inch even at 1000 yards, this error compounds significantly on longer range shots that may require adjustment upwards of 20–30 MOA to compensate for the bullet drop. If a shot requires an adjustment of 20 MOA or more, the difference between true MOA and SMOA will add up to 1 inch or more. In competitive target shooting, this might mean the difference between a hit and a miss.
The physical group size equivalent to m minutes of arc can be calculated as follows: group size = tan(m/60) × distance. In the example previously given, for 1 minute of arc, and substituting 3,600 inches for 100 yards, 3,600 tan(1/60) ≈ 1.047 inches. In metric units 1 MOA at 100 metres ≈ 2.908 centimetres.
Sometimes, a precision-oriented firearm's performance will be measured in MOA. This simply means that under ideal conditions (i.e. no wind, high-grade ammo, clean barrel, and a stable mounting platform such as a vise or a benchrest used to eliminate shooter error), the gun is capable of producing a group of shots whose center points (center-to-center) fit into a circle, the average diameter of circles in several groups can be subtended by that amount of arc. For example, a 1 MOA rifle should be capable, under ideal conditions, of repeatably shooting 1-inch groups at 100 yards. Most higher-end rifles are warrantied by their manufacturer to shoot under a given MOA threshold (typically 1 MOA or better) with specific ammunition and no error on the shooter's part. For example, Remington's M24 Sniper Weapon System is required to shoot 0.8 MOA or better, or be rejected from sale by quality control.
Rifle manufacturers and gun magazines often refer to this capability as sub-MOA, meaning a gun consistently shooting groups under 1 MOA. This means that a single group of 3 to 5 shots at 100 yards, or the average of several groups, will measure less than 1 MOA between the two furthest shots in the group, i.e. all shots fall within 1 MOA. If larger samples are taken (i.e., more shots per group) then group size typically increases, however this will ultimately average out. If a rifle was truly a 1 MOA rifle, it would be just as likely that two consecutive shots land exactly on top of each other as that they land 1 MOA apart. For 5-shot groups, based on 95% confidence, a rifle that normally shoots 1 MOA can be expected to shoot groups between 0.58 MOA and 1.47 MOA, although the majority of these groups will be under 1 MOA. What this means in practice is if a rifle that shoots 1-inch groups on average at 100 yards shoots a group measuring 0.7 inches followed by a group that is 1.3 inches, this is not statistically abnormal.
The metric system counterpart of the MOA is the milliradian (mrad or 'mil'), being equal to 1⁄1000 of the target range, laid out on a circle that has the observer as centre and the target range as radius. The number of milliradians on a full such circle therefore always is equal to 2 × π × 1000, regardless the target range. Therefore, 1 MOA ≈ 0.2909 mrad. This means that an object which spans 1 mrad on the reticle is at a range that is in metres equal to the object's size in millimetres[dubious – discuss] (e.g. an object of 100 mm subtending 1 mrad is 100 metres away). So there is no conversion factor required, contrary to the MOA system. A reticle with markings (hashes or dots) spaced with a one mrad apart (or a fraction of a mrad) are collectively called a mrad reticle. If the markings are round they are called mil-dots.
