It's simpler than trying to figure MOA, which is inch-based.
To be accurate, MOA is a pure angular measurement which isn't based on any particular linear measurement. Angular measurements can be related to linear measurements using mathematical relationships, but they don't automatically relate to any particular linear measurement unit until someone chooses to do the math to relate them to one they choose.
In other words, MOA is no more closely related to inches than it is to mm. It's just that the fact that 1MOA is conveniently close to 1" at 100 yards that makes people think of it most often in terms of inches.
In reality, one could just as well think about 1MOA being about 30mm at 103 meters if they wished.
Mils, in theory relate to native angular measurement--radians which are expressed in terms of pi, rather than a relatively arbitrary number of divisions of a circle. So there are 2 pi radians in a circle which is equivalent to 360 degrees or 21,600 minutes. Mils are actually milliradians--or a thousandth of a radian. At least in theory. In practice, "convenient" values have been chosen. If you really calculate things out, there would be about 6283.2 milliradians (Mils) in a circle, but for convenience, the value actually used for calculations is a more round number. NATO's milliradian is 1/6400th of a circle, but the Warsaw Pact used 1/6000th of a circle as their milliradian and there are some other values also floating around out there as well.
Exact trig value below was calculated to 102 decimal places for those wanting more precision with trigonometry.
Approximating 1MOA as 1" at 100 yards is the same thing as saying that pi is equal to 3. No need to go out to 102 decimal places to see that they are obviously different.
Calling 1MOA 1" @ 100 yards is fine as a convention for the sake of simplicity--I've done it myself on many an occasion. But it's good for people to understand the true origin of the term, and once they do, it's also good for them to understand that the common usage isn't equivalent to the mathematical usage of the identical term.
Even though a minute of angle is a unit of angular measurement that cannot possibly magically turn itself into an inch, or any other unit of linear measurement, we all use MOA in other ways such as "a sub-MOA group", "a 2 MOA shooter", "a 1 MOA rifle" etc, which is fine.
Arc lengths are commonly measured using typical "linear" measurements with no issues at all--one does need to keep in mind whether the length is curved or straight, but as long as that is done, inches can be used to measure distances on a straight line or a curved one.
You are correct that any angular measurement, when used to measure distance, conventionally refers to an arc length rather than a straight line unless it is specified otherwise. However, there's nothing especially problematic with drawing a straight line from one end of the arc to the other and measuring that distance (which is called a chord) instead of the arc length.
It turns out that when the angle is small, the length of the chord is very close to the length of the arc for exactly the same reason that the
sine of an angle measured in radians is approximately equal to the angle itself when the angle is small. When one measures an angle in minutes (21,600ths of a circle), the angles tend to be very small indeed which results in the difference of the arc length and the chord length being very small.
Not in the sense that people mean when they bristle at those who accurately point out that 1MOA is actually about 5% larger than 1" at 100 yards--but really, really tiny.
For 1MOA, for example, the difference between the chord length (linear distance on a flat target) vs the arc length at 100 yards is about 0.0000000037 inches. About 750,000 times less than the thickness of a human hair.
It's definitely interesting and useful to keep all the terms straight, but people who get bent out of shape when someone notes a difference of about 5% between an approximation/convention and reality are going to have their heads explode when someone starts talking about differences measured in hundred-millionths of an inch.
Group MOA = Group spread (in inches)/{36*[target distance (in yards)]*tan(1/60)}
For an easy to remember approximation of 1MOA, take the distance to the target in yards, divide by 95.5 and that's about how big 1MOA is at that distance. The error will be less than 0.0075% from the actual value.
Using the same general idea, one can get a pretty accurate group size in MOA by multiplying the group size in inches by 95.5 and dividing by the distance to the target in yards.
To get closer to the true value of 1MOA than the approximation above provides, one can multiply the distance to the target in yards by pi and divide by 300.
Using that approach, one can find the group size in MOA by multiplying the group size in inches by 300, dividing by pi and then by the distance to the target in yards.
If you believe that 3 is a good approximation for pi, then that just works out to one inch at 100 yards.
