Here is a guide to understanding minutes of angle (MOA), what is truly meant when discussing accuracy in the context of MOA, and some other useful bits such as mils and estimating range. I see a lot of people throwing around the term MOA. Many times people don't understand what they are talking about and if they do, they do not understand where the term comes from.
This post is going to be similar to a PDF document I read years ago. Some of you may notice the similarity. For those that found such documents confusing, hopefully a second opinion on this topic will provide clarity.
So, what the heck are mils? What is this MOA thingy? Why is this useful information for marksmen?Mils
The term mil is a shortened term for milliradian. This is a unit of measure for a segment of a circle. You may remember from trigonometry that a circle has 360 degrees. Milliradians are a similar measurement but much more precise. A circle, no matter how big, consists of 6,283.2 milliradians. A mil is a unit of angular measure just like a degree. It subtends an arc of the circle.
If you start with a point in the center of a circle and extend two lines to the edge of the circle, a segment or arc of the circle is created inside of the angle formed by the two lines. The length of the arc is what we are measuring whether we use degrees, mils, or minutes. You may have seen an optic reticle that has "mil dots" on it like this one:
Those little dots on the crosshairs are called mil dots for good reasons that we will soon understand. They are used to estimate the distance to a target which is vertically used to calculate hold over, and used horizontally to adjust for windage or movement.MOA
MOA or minutes of angle are similar to degrees or mils. Minutes in this context has nothing to do with time. It is another way to measure a piece or arc of a circle. It is even more precise than degrees or mils. There are 21,600 minutes in a circle. MOA is also used to discuss shot groupings although generally using the 1 inch at 100yards equals 1 MOA formula. Many people never make it beyond this definition and for good reason. It is a very close explanation and can be considered a practical measurement. Many refer to this rule of thumb as the "shooter's MOA". I like to call it fuzzy MOA and while I use it myself, many people learn this tidbit of info and then act as if they have the golden ticket. It is however, not exactly correct no matter how practical it is.Radians and Mils in a Circle
Let's back up for a second. Some more explanation is in order to help us piece all of this together. Let's consider something called a radian. It is a unit of measure just like degrees, mils, and MOA. It is the most imprecise of all of these. If you are following this so far, that means a radian is larger than a degree thus a bigger arc or segment of the circle is inside an angle of 1 radian. If you know anything about the metric system and prefixes, you probably guessed that a radian is 1000 milliradians.
The technical definition of a radian is "the angle that subtends an arc equal in length to the radius of a circle". A picture will explain this wordy definition:
You can see the radius and the arc are both labeled "1" and are equal in length. The angle that is required to form an arc the same length as the radius is 1 radian of measure. The angle is called a radian angle or just radian and the arc is called a radian arc. That picture should also show why there are the same number of radians in every circle. If the radius is bigger or smaller, so is a radian arc of that circle since both are equal in length.
So, how many radians are in a circle? Well, there are the same amount of radian angles or radians as there are radian arcs. How can we tell how many radian arcs there are? Easy, just find the circumference of the circle and divide that by the radius of the circle. It's like pieces of a pie and I know you folks like pie.
Instead of talking about pie, lets talk about pi (π) or 3.14159. The circumference of a circle is measured with this formula: C=2πr
A minute ago I stated that the number of radians always equals the number of radian arcs. This allows us to take the circumference and divide it by the radius length to find how many radians are in the circle. Since we know π = 3.14159 and every circle has the same number of radians, we can simply do 2πr/r = 2π = 2*3.14159. This equals 6.2832 radians or 6,283.2 milliradians.Minutes in a Circle
The number of minutes in a circle is more of a loose definition. There are 60 minutes or minutes of angle in 1 degree of a circle. Since we all know there are 360 degrees in a complete circle, then there are 360*60 or 21,600 minutes of angle in every circle.Minutes per Milliradian
You might wonder how many minutes are in 1 mil. Easy! Just take the number of minutes in a circle divided by the number of mils.
21,600/6,283.2 = 3.438 minutes per mil.Degrees per Milliradian
Since we have done all the work this is another easy one. Take the number of degrees in a circle and divide by the number of mils.
360/6,283.2 = 0.0573 degrees per mil.Inches per Mil at 100 yards
Building on what we have done so far, lets calculate something more practical for marksmen and find out how many inches are in a mil at 100 yards. We know that the radius of a circle is equal to a radian of arc. Therefore a mil of arc is equal to 1/1000th of radius, right? So taking 100 yards or 3600 inches (100y = 300ft = 3600in) if we find 1/1000th of 100 yards, that gives us 1 mil of 100 yards. To find this in inches we take 3600 inches divided by 1000 milliradians.
3600/1000 = 3.6 inches per mil at 100 yards.
Imagine the line drawn through your scope to the target is the radius of your circle. The height of 1 mil in your scope is going to be 3.6 inches if your target is 100 yards away.Inches per Minute at 100 yards
We are approaching the reason why fuzzy MOA or the definition of MOA in general can be tossed around as 1" at 100 yards. Lets find out why.
Remember before we found that there are 3.6 inches per mil. Also, we know that there are 3.438 minutes per mil. If we divide these numbers we can find the value of inches per minute.
3.6/3.438 = 1.047 inches per minute at 100 yards.
Ahh, so the exact definition of MOA is revealed. 1.047 inches at 100 yards. That extra 0.047 inches is tossed out the window when we discuss MOA in most contexts. For practical reasons, it is not enough to make a significant difference.Range Estimation using Mil Dots
More work can be done to derive the actual math behind range estimation with mil dots. While useful knowledge, that is beyond the scope of this post (I am tired of typing). For now I will leave you with these handy formulas that you can use to estimate range if you know the size of your target and are looking through your glass that has a mil dot reticle:
(height of target in yards / mils) * 1000 = distance to target in yards
(height of target in inches / mils) * 27.78 = distance to target in yards
(height of target in inches / moa) * 100 = distance to target in yardsSummary
At this point, either you skipped this thread, your head is hurting, or you've got a light bulb floating above it. As I mentioned before, I am as guilty of using fuzzy MOA as anyone when talking about accuracy or group size. I hope this sheds some light on the real ins and outs of MOA.