Before writing this, I performed a quick 'hide google search. Although this topic has been beaten to death, I couldn't find a topic dedicated to explaining the math behind mils and MOAs. I saw many explanations and cheat sheets pertaining to practical matters and practice, but no real detailed explanations. I apologize if there is indeed a section that covers this in the detail I am about to provide, but I just didn't see it, even in the FAQ section
All too often, I notice that people are quite perplexed about the understanding of what a mil and an MOA actually is. This is understandable to an extent, since odd conversion numbers are thrown around on a regular basis. Most people simply write down or memorize the numbers, but never actually take the time to work out the math. In doing so, one will have a complete understanding of what the numbers mean. Note that one need not be a mathematician, just a basic understanding of simple geometry and a calculator with trigonometric functions. Here I will show what mils and MOAs are and how to convert between them in detail. I used yards for my examples, but the same concepts apply to meters. If meters are desired, they can be calculated by performing some simple conversions:
<span style="font-weight: bold"><span style="font-style: italic">Pi</span></span>: What is <span style="font-style: italic">pi</span>? Most people blurt out "3.14", as they have been drilled in elementary school. But we need to understand what <span style="font-style: italic">pi</span> actually represents to move on. <span style="font-style: italic">Pi</span> is the multiple of diameter lengths of a circle equal to its circumference. So in a circle whose diameter (d) equals 1, its circumference (C) equals <span style="font-style: italic">pi</span>:
<span style="font-weight: bold">C = <span style="font-style: italic">pi</span> * d </span>
d = 1, so C = <span style="font-style: italic">pi</span>
So if you roll a hoop with a 1 foot diameter through one complete revolution, it will travel
~ 3.14 feet on the floor, denoting its circumference. The actual value of <span style="font-style: italic">pi</span> is a never ending non repeating decimal. On my calculator, it goes out to 3.141592653589793. It is not necessary to extend the number out past the hundredths (3.14) for simple calculations, but for the calculations here, I will use the full number provided by my calculator to get the most precise values.
<span style="font-weight: bold">Radians</span>: Most people could not tell you what a radian is. In a circle, 1 radian equals the <span style="text-decoration: underline">angle</span> formed by the arc length, when the arc length is equal to the radius (r) of the circle. The arc length is like a side of a triangle, but the side is curved so we cannot use simple trigonometry here. We know that C = <span style="font-style: italic">pi</span>*d, so C = 2<span style="font-style: italic">pi</span>*r. There are 360 degrees in a circle, so the length of 360 degrees C equals 2<span style="font-style: italic">pi</span>*r/r = 2<span style="font-style: italic">pi</span> radians. So 360 degrees equals 2<span style="font-style: italic">pi</span> radians; expressed another way, 1 radian = 180/<span style="font-style: italic">pi</span> degrees. This yields <span style="font-weight: bold">~ 57.3 degrees</span>. So there are 2<span style="font-style: italic">pi</span> radians or ~ 6.28 radians in a circle (360/2<span style="font-style: italic">pi</span>).
<span style="font-weight: bold">Milliradians</span>: A milliradian or mil is simply what it sounds like; 1/1000 of a radian. This is equal to <span style="font-weight: bold">0.0573 degrees</span>. Now comes the trigonometry. Take a known distance; any distance you please. For this example, we will use 100 yards. Now form a triangle. 100 yards is the length of the triangle, and we want to find the height. To do so, we take the tangent of the angle, and multiply it by the length. We then perform dimensional analysis for unit conversion of yards into inches. A mil is an angle, so it is represented by the Greek letter <span style="font-style: italic">theta</span>:
tan(<span style="font-style: italic">theta</span>) = y/x
tan(0.0573) = y/100 yards
y = 0.001 * 100 yards = 0.1 yards * 36 inches/yard = <span style="font-weight: bold">3.6 inches</span>
So via simple trigonometry, we see that <span style="font-weight: bold">1 milliradian = 3.6" at 100 yards</span>. Note that <span style="text-decoration: underline">mils have nothing to do with the metric system</span>; as stated above, a mil is just what is sounds like; a thousandth of a value. In this case, that value is a radian, and a thousandth of a radian equals the angle that subtends the height y opposite of that angle.
<span style="font-weight: bold">Minute of Angle (MOA)</span>: An MOA is very simple to understand. Like a clock, minutes and seconds are 1/60 of hours and minutes, respectively. The angle in question is 1 degree. So an MOA is 1/60 degrees, or <span style="font-weight: bold">0.01667 degrees</span>. We perform the same trigonometry as above to find y. At 100 yards:
tan(<span style="font-style: italic">theta</span>) = y/x
tan(0.01667) = y/100 yards
y = 0.000291 * 100 yards = 0.0291 yards * 36 inches/yards = <span style="font-weight: bold">1.0472 inches</span>
So <span style="font-weight: bold">1 MOA equals 1.0472" at 100 yards</span>. MOAs should not be confused with inches per hundred yards (IPHY). Yes, for ease we can round down and say that 1 MOA = 1" at 100 yards, but this is for estimation only. At long distances, it may be necessary to multiply the value by 95% to get a more precise value.
Now comes the "difficult" part. How do we convert from mils to MOA? Actually, this is not so difficult if you went through the trouble to find the values above. We found that at 100 yards, 1 mil = 3.6", and 1 MOA = 1.0472". So we perform some simple math and dimensional analysis:
3.6 inches/mil * 1 MOA/1.0472 inches = <span style="font-weight: bold">3.4377 MOA/mil</span>
We all know the equation for ranging in yards:
(target in inches * 27.778)/mils = distance in yards
But where does the 27.778 come from? That is a conversion that can be calculated by plugging in known values calculated above and solving for x:
(3.6 inches * x)/1 mil = 100 yards
x = (100 yards * 1 mil)/3.6 inches = 27.778 yards * mil/inch
Note that the units are confusing in this solution. This value denotes that at a distance of 27.778 yards, a 1" target is 1 milliradian. When these units are incorporated into the original equation, the units all cancel out, leaving yards on the right side of the equal sign and will give us the distance which we are looking for.
In closing, I want to reiterate a few points:
As stated above, mils are not metric; they are a simple shift of 3 decimal places to the left. Yes, at 100 meters, a 0.1 mil adjustment of the dial will shift the POI by 1 cm, but this is only because you are moving the decimal place without changing the units. At 100 yards, a 0.1 mil adjustment moves the POI by 0.01 yards or 0.36". In essence, a 0.1 mil adjustment moves the POI by 1 "centiyard". That term is technically correct, but in actuality it is slang, and only yields 127 results in a search on google.
On the surface, MOA seems easier to use than mils because of the misconception that MOA reads in inches and mils read in cm. However, most confusion arises because of the traditional mil reticle with MOA dials. The MOA scopes are indeed easier to use than a traditional mildot scope with MOA dials, but this is only because most, if not all MOA scopes come with matching turrets. In actuality, it is much easier to simply move the decimal place, which mils allow you to do. I am not advocating mil based reticles over MOA; I have 3 NF scopes with MOA reticles and I like them very much. Just keep in mind that when considering a new scope and are deciding between a mil or MOA reticle, be sure to get matching turrets; you will have a much easier time than with the traditional mil reticle/MOA turret system and neither choice is a compromise. It is a matter of personal preference.
I hope this helps.
-mike
All too often, I notice that people are quite perplexed about the understanding of what a mil and an MOA actually is. This is understandable to an extent, since odd conversion numbers are thrown around on a regular basis. Most people simply write down or memorize the numbers, but never actually take the time to work out the math. In doing so, one will have a complete understanding of what the numbers mean. Note that one need not be a mathematician, just a basic understanding of simple geometry and a calculator with trigonometric functions. Here I will show what mils and MOAs are and how to convert between them in detail. I used yards for my examples, but the same concepts apply to meters. If meters are desired, they can be calculated by performing some simple conversions:
<span style="font-weight: bold"><span style="font-style: italic">Pi</span></span>: What is <span style="font-style: italic">pi</span>? Most people blurt out "3.14", as they have been drilled in elementary school. But we need to understand what <span style="font-style: italic">pi</span> actually represents to move on. <span style="font-style: italic">Pi</span> is the multiple of diameter lengths of a circle equal to its circumference. So in a circle whose diameter (d) equals 1, its circumference (C) equals <span style="font-style: italic">pi</span>:
<span style="font-weight: bold">C = <span style="font-style: italic">pi</span> * d </span>
d = 1, so C = <span style="font-style: italic">pi</span>
So if you roll a hoop with a 1 foot diameter through one complete revolution, it will travel
~ 3.14 feet on the floor, denoting its circumference. The actual value of <span style="font-style: italic">pi</span> is a never ending non repeating decimal. On my calculator, it goes out to 3.141592653589793. It is not necessary to extend the number out past the hundredths (3.14) for simple calculations, but for the calculations here, I will use the full number provided by my calculator to get the most precise values.
<span style="font-weight: bold">Radians</span>: Most people could not tell you what a radian is. In a circle, 1 radian equals the <span style="text-decoration: underline">angle</span> formed by the arc length, when the arc length is equal to the radius (r) of the circle. The arc length is like a side of a triangle, but the side is curved so we cannot use simple trigonometry here. We know that C = <span style="font-style: italic">pi</span>*d, so C = 2<span style="font-style: italic">pi</span>*r. There are 360 degrees in a circle, so the length of 360 degrees C equals 2<span style="font-style: italic">pi</span>*r/r = 2<span style="font-style: italic">pi</span> radians. So 360 degrees equals 2<span style="font-style: italic">pi</span> radians; expressed another way, 1 radian = 180/<span style="font-style: italic">pi</span> degrees. This yields <span style="font-weight: bold">~ 57.3 degrees</span>. So there are 2<span style="font-style: italic">pi</span> radians or ~ 6.28 radians in a circle (360/2<span style="font-style: italic">pi</span>).
<span style="font-weight: bold">Milliradians</span>: A milliradian or mil is simply what it sounds like; 1/1000 of a radian. This is equal to <span style="font-weight: bold">0.0573 degrees</span>. Now comes the trigonometry. Take a known distance; any distance you please. For this example, we will use 100 yards. Now form a triangle. 100 yards is the length of the triangle, and we want to find the height. To do so, we take the tangent of the angle, and multiply it by the length. We then perform dimensional analysis for unit conversion of yards into inches. A mil is an angle, so it is represented by the Greek letter <span style="font-style: italic">theta</span>:
tan(<span style="font-style: italic">theta</span>) = y/x
tan(0.0573) = y/100 yards
y = 0.001 * 100 yards = 0.1 yards * 36 inches/yard = <span style="font-weight: bold">3.6 inches</span>
So via simple trigonometry, we see that <span style="font-weight: bold">1 milliradian = 3.6" at 100 yards</span>. Note that <span style="text-decoration: underline">mils have nothing to do with the metric system</span>; as stated above, a mil is just what is sounds like; a thousandth of a value. In this case, that value is a radian, and a thousandth of a radian equals the angle that subtends the height y opposite of that angle.
<span style="font-weight: bold">Minute of Angle (MOA)</span>: An MOA is very simple to understand. Like a clock, minutes and seconds are 1/60 of hours and minutes, respectively. The angle in question is 1 degree. So an MOA is 1/60 degrees, or <span style="font-weight: bold">0.01667 degrees</span>. We perform the same trigonometry as above to find y. At 100 yards:
tan(<span style="font-style: italic">theta</span>) = y/x
tan(0.01667) = y/100 yards
y = 0.000291 * 100 yards = 0.0291 yards * 36 inches/yards = <span style="font-weight: bold">1.0472 inches</span>
So <span style="font-weight: bold">1 MOA equals 1.0472" at 100 yards</span>. MOAs should not be confused with inches per hundred yards (IPHY). Yes, for ease we can round down and say that 1 MOA = 1" at 100 yards, but this is for estimation only. At long distances, it may be necessary to multiply the value by 95% to get a more precise value.
Now comes the "difficult" part. How do we convert from mils to MOA? Actually, this is not so difficult if you went through the trouble to find the values above. We found that at 100 yards, 1 mil = 3.6", and 1 MOA = 1.0472". So we perform some simple math and dimensional analysis:
3.6 inches/mil * 1 MOA/1.0472 inches = <span style="font-weight: bold">3.4377 MOA/mil</span>
We all know the equation for ranging in yards:
(target in inches * 27.778)/mils = distance in yards
But where does the 27.778 come from? That is a conversion that can be calculated by plugging in known values calculated above and solving for x:
(3.6 inches * x)/1 mil = 100 yards
x = (100 yards * 1 mil)/3.6 inches = 27.778 yards * mil/inch
Note that the units are confusing in this solution. This value denotes that at a distance of 27.778 yards, a 1" target is 1 milliradian. When these units are incorporated into the original equation, the units all cancel out, leaving yards on the right side of the equal sign and will give us the distance which we are looking for.
In closing, I want to reiterate a few points:
As stated above, mils are not metric; they are a simple shift of 3 decimal places to the left. Yes, at 100 meters, a 0.1 mil adjustment of the dial will shift the POI by 1 cm, but this is only because you are moving the decimal place without changing the units. At 100 yards, a 0.1 mil adjustment moves the POI by 0.01 yards or 0.36". In essence, a 0.1 mil adjustment moves the POI by 1 "centiyard". That term is technically correct, but in actuality it is slang, and only yields 127 results in a search on google.
On the surface, MOA seems easier to use than mils because of the misconception that MOA reads in inches and mils read in cm. However, most confusion arises because of the traditional mil reticle with MOA dials. The MOA scopes are indeed easier to use than a traditional mildot scope with MOA dials, but this is only because most, if not all MOA scopes come with matching turrets. In actuality, it is much easier to simply move the decimal place, which mils allow you to do. I am not advocating mil based reticles over MOA; I have 3 NF scopes with MOA reticles and I like them very much. Just keep in mind that when considering a new scope and are deciding between a mil or MOA reticle, be sure to get matching turrets; you will have a much easier time than with the traditional mil reticle/MOA turret system and neither choice is a compromise. It is a matter of personal preference.
I hope this helps.
-mike
