A Shooters Guide to Understanding Mils and MOA and their range Estimation Equations By Robert J. Simeone
Shooters of all types, from military and police, to hunters and target shooters, use milliradians (mils) and minutes of angle (moa) to estimate the distance to their targets and to adjust their scopes in order to get their bullets on the intended point of impact. They also use them to adjust shots for winds and the movements of targets. Also, shooters frequently talk about their shot groupings in terms of moa, such as “my rifle shoots one moa all the time”. Even though many shooters use mils and moa, there is still a lot of confusion and misunderstanding of exactly what they are and where they come from. Therefore, I’m going to try and explain them in simple terms and clear up any confusion you might have of these tools we so often use in precision shooting. Let’s start with the basics:
The Angles You might remember the term “radians” from that trigonometry class you took in high school. In case you forgot, let’s review. A radian is an angular measurement like degrees, but unlike degrees, is based on the actual physical properties of a circle. Officially, one radian subtends an arc equal in length to the radius (r) of a circle. If that didn’t clear things up, then, try this. If you take the radius of a circle and superimpose its length on the circumference of the circle, you create an “arc” with a length equal to the radius of the circle. Now, connect both ends of this “radius arc” to the center, and the angle created by the three sides equals 1 radian (see figure below).
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To find out how many radians (the angles at the center) are in a circle, we use simple math. Remember, the “radian angle” at the center is directly related to the “radius” that we artificially superimposed on the circumference of the circle. Therefore, if we find out how many “radii” can fit around the circumference, we’ll then also know the number of “radians” (the angles at the center) there are in a circle. To do this, we use the circumference formula of a circle, which is C = 2 r. Take 2 r and divide by “r”. (We divide by “r” because that will give us the number of radii that can go around the circumference of a circle, remembering that radian angles at the center are equal to this number). (Note:
= 3.14159…...)
2 r=2 r=2 r r
= 2 x 3.14159 = 6.2832.
Therefore, there are 6.2832 radians in a circle (and for that matter, 6.2832 radii that can go around the circumference of a circle). No matter how long the radius “r” is, there will always be 6.2832 radians in any size circle because the “r” always gets cancelled out in the math (see above) and all you’re left with is 2 . To get an idea of how big one radian is, we can convert it to the more familiar degrees. To do that, take the 360° that every circle has and divide that by the 6.2832 radians that every circle also has and you get 57.3 degrees per radian. So then, what’s a “milliradian”? “Milli” is by definition, 1/1000th. Therefore, a “milliradian”, shortened to “mil” by shooters, is 1/1000 of a radian. Therefore, take each of the 6.2832 radians in a circle and divide each one into 1000 smaller angles. When you do that you’ll get 6.2832 x 1000 = 6283.2 milliradians in every circle. Since there are 6283.2 milliradian in a circle compared to 360°, we have a finer angle of measurement than degrees.
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Note: There is some controversy about what type of “mils” American military and tactical shooters use. Some think they use a mil that is based on a circle that has artificially been divided into 6400 mils instead of 6283.2. This has widely been circulated, written about and even taught, including in the military. But this is not the case. While it may be true that some artillery and other military units do use this type of mil (the one based on 6400 mils), American military snipers and tactical shooters use scopes that are based on and calibrated using “real mathematical milliradians”, which is 6283.2 milliradians (mils) per circle. It should be noted that some other countries, like Russia for example, do use different values for their mils on their scopes, but for American shooters, our “mil” is 6283.2 milliradians.
The other angle shooters use is “minute of angle”. (Note: “minute of angle” is interchangeable with and is the same thing as “minutes of angle”, or “moa”). Moa is a little easier to understand than mils since it is a subset of the more familiar degrees. Every circle has 360 degrees in it. Each one degree is further divided into 60 minutes (or 1° = 1 moa). Therefore: 60 360 (degrees) x 60 (minutes) = 21,600 minutes in every circle (or 21,600 moa). 1 (degree) This is an even more precise unit of angular measurement than mils (21,600 moa vs. 6283.2 mils). At this point you might ask, “How come shooters use two different angular measurements?” In short, for years, rifle scopes mostly used moa increments on their target knobs for shot adjustments. Much later on the military come up with the mil-dot reticle for snipers to estimate range, but the scopes still kept their turret adjustments in moa. So you had the “mil” angular system for range estimation, and the “minute” angular system for shot adjustment. It can get confusing converting between the two and for many shooters it doesn’t make a lot of sense. Fortunately, many scopes come with the same system for both range estimation and shot adjustments, which I recommend for several reasons, one being that you don’t have to convert between the two different angles. Now that we know what a mil and moa are and where they come from, how can shooters use these? The answer is simple. With known angles (mils or moa) and lengths (heights of objects), we can compute distances to targets using formulas which are based on trigonometry, which is the math of right triangles. But before we get into that, let’s talk about a few common conversions and relations. We’ll start with the conversion between mils and moa. This is important because as stated above, many scopes use reticles etched in mils but have there turret adjustments in moa. Since we know from above that we have 21,600 minutes and 6283.2 mils in every circle, the conversion between the two is easy to figure out. Take 21,600 minutes and divide that by 6283.2 mils and you get: 21600 minutes = 3.438 minutes per mil. 6283.2 mils
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Therefore, the two different “angles” can be compared by saying 1 mil equals 3.438 moa. Next is the common benchmark shooters use, the 100 yard shot. At that distance, it would be nice to know the size in inches that the angle of one mil or one moa gives you. Look at the figure below. (Try to remember that this is really just a small section of a much larger circle that has a radius of 100 yards).
We have an angle, 1 mil, and a distance down range, 100 yards (which is also 3,600 inches). This angle and distance forms a triangle. Our goal is to find out what the size of “x”, the opposite side, is. Or put another way, what height does the angle of 1 mil give us at 100 yards distance. Math has supplied us with a nice tool to get that answer, the tangent function, which is the ratio of the opposite side to the adjacent side of a right triangle. To use the tangent function, it’s easier if we convert the angle that is expressed in “mils” into an angle that is expressed in “degrees”. Similar to what we did earlier, but this time dividing 360° by 6283.2 mils, you’ll get the number of degrees in every mil. 360°/6283.2 mils = .0573° per mil. Now break out the calculator and we can solve for “x”, the height that 1 mil at 100 yards equals: 1) Tan θ = opposite adjacent
2) Tan (.0573°) = 1 mil ↑
x 3) 3,600 (Tan .0573) = x 3,600 in.
(Note: using your calculator, the tangent of .0573 = .001)
4) 3,600 (.001) = x
5) x = 3.6 inches
That’s where the familiar “1 mil at 100 yards equals 3.6 inches in height” comes from. It’s always been just simple trigonometry. Note: Even though the opposite side of the triangle, “x” in Fig.5 above, is really not a straight line but a curve because it is actually a part of a much larger circle, it is a very small curve. At this distance and at this small of an angle, for all practical purposes we can consider it a straight line and the curve’s effects on the math are negligible.
How about 1 moa at 100 yards? Recall that 1 minute (or 1 moa) equals 1/60th of a degree: 1° = .016667°. 60
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Again, using the tangent function and solving for the opposite side “x”: 1) Tan θ = opposite adjacent
2) Tan (.016667°) = 1 moa ↑
x 3) 3,600 (Tan .016667) = x 3,600 in.
Note: On the calculator, the tangent of .016667° = .00029089.
4) 3,600 (.00029089) = x
5) x = 1.047 inches
Therefore, the angle of 1 moa at 100 yards equals 1.047 inches in height. Having 1.047 inches at 100 yards is really nice since it can easily be rounded to 1 inch, something we can comprehend and visualize easily. There is a third angular measurement that many shooters use and that many shooters don’t even know they have on some of their scopes. Guessing simplicity being the reason, shooters wanted another angle that would give them “exactly” 1 inch at 100 yards. This angular measurement is known as “inch per 100 yards”, or more simply, “shooters moa” (s-moa). It is close to the actual “true” moa of 1.047 inches at 100 yards. Some reticles are etched in this measurement, and some turret adjustments, even though they say “moa”, are actually calibrated in s-moa instead of “true” moa. The math is similar to what we just did above, except this time we first define the lengths we want (100 yards distance and “1” inch in height), then solve for the angle that will give us these measurements.
To do this we use the “inverse” tangent of trigonometry to solve for the angle: 1) Tan θ = opposite adjacent
2) Tan θ = 1 s-moa ↑
1 3) Tan (θ) = .000277778 3,600 inches.
Note: On the calculator, the “inverse tangent” of .000277778 = .0159154°.
Therefore, 1 s-moa equals .0159154°. Very close to what the actual “true” moa equals, .016667°.
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So let’s review. This is what we have while looking through a mil dot rifle scope at 100 yards. Each dot is separated by 1 mil:
Keep in mind that when we talk about “mils”, we’re talking about the angular measurement of a circle in “milliradians”. When we talk about “moa”, we’re talking about the angular measurement of a circle in “minutes”, which are a subset of degrees. They’re two different angular measurements, but measuring the same thing, which in our case are really just small sections of a circle that look like triangles like in the figures above.
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The Range Estimation Equations Using all this information and applying trigonometry and algebra, we can come up with simple range estimation equations that we use to estimate the distance of objects of known height without resorting to trigonometry and using calculators. (The math unfortunately is to long for this article, but if you really want to see it, you can go to: http://files.thetallengineer.com/RangeEstimation_Rev1.pdf The basic range estimation equations are. Height of Target (yards) x 1000 = Distance to Target (yards) mils Height of Target (inches) x 27.78 = Distance to Target (yards) mils Height of Target (inches) x 95.5 = Distance to Target (yards) moa Height of Target (inches) x 100 = Distance to Target (yards) s- moa
Using the Range Estimation Equations:
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In short, that’s the background of mils, moa and s-moa, and their range estimation equations. I hope this article gives you a little better understanding of the angles and terms we so often use in precision shooting.
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MIL/MOA/S-MOA Distance Equations Below are the distance equations for various combinations of reticles, units of height, and units of distance to the target. Choose the one that you have a reticle for and that has the units of measurement you prefer. Height of Target (yards) x 1000 = Distance to Target (yards) mils Height of Target (inches) x 27.78 = Distance to Target (yards) mils Height of Target (inches) x 25.4 = Distance to Target (meters) mils Height of Target (meters) x 1000 = Distance to Target (meters) mils Height of Target (cm) x 10 = Distance to Target (meters) mils Height of Target (inches) x 95.5 = Distance to Target (yards) moa Height of Target (inches) x 87.32 = Distance to Target (meters) moa Height of Target (meters) x 3437.75 = Distance to Target (meters) moa Height of Target (cm) x 34.37 = Distance to Target (meters) moa Height of Target (inches) x 100 = Distance to Target (yards) s- moa Height of Target (inches) x 91.44 = Distance to Target (meters) s- moa Height of Target (meters) x 3600 = Distance to Target (meters) s- moa Height of Target (cm) x 36 = Distance to Target (meters) s- moa
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