As popular and mainstream as long range shooting has become in the years since I started building rifles, I still sense confusion from my new customers and clients over some of the basic concepts involved. I believe this is partially due to several names and acronyms that get passed around for the same subject. In this article, that subject is angular measurements which are used to measure or brag about a rifle's accuracy as well as the units on what happens in the guts of your riflescope when you start cranking turrets.

The concept of measuring an angle might cause flashbacks to your high school geometry class, but the basic math involved is necessary to manipulate a rifle and scope to shoot long range. The math involved is nothing more than addition and subtraction of angular measurements. Those angular measurements just have units that tend to draw confusion. It is my goal that this article clears that up for the shooters. We will dive right into using angular measurements to manipulate your rifle scope. Using angular measurements to discuss accuracy should be elementary once we get refreshed on the concept of angular measurements.

When you throw a football, you are launching a projectile at some angle relative to your line of sight. You look at your receiver, but instinct reminds you about gravity,and you somehow calculate a trajectory in your head and release the football at an angle that sets the trajectory for the pass. I am baffled at how effective the human can be at this task without even scratching numbers into the dirt. As a shooter we are doing the same thing. We do have higher standards for accuracy and most likely are attempting to connect with the receiver at much further distances. We have the ability to measure things like our bullet's speed and its characteristics of flight, and the distance to our target with great certainty. Somewhat recent advancements in optics and ballistics calculators let us precisely account for gravity even at budget prices. As technical as all this sounds, all we are doing with the rifle scope to compensate for gravity's effect on the projectile over our shot distance is adjusting the angle between the line of sight and the bore angle. Because the human gets to talk during this adjustment, things can still get messy.

"What was your come-up?"
"7.4 Mils"
"What is that in minutes?"
"I don't know, just come down 2 clicks"
"Impact"

Going back to geometry class for a minute, an angular measurement is best shown as the arc of a fraction of the circumference of a circle, the crust on a piece of the pie. In shooting, that slice becomes very thin which comes into play as the small-angle approximation (more on that in a bit). There are two units of measurements that we use to measure angles, or arcs of a circle, degrees and radians. A full circle encompasses 360 degrees, which is knowledge most people have an understanding of. That circle also has 2 x Pi radians of circumference, where Pi is a number that relates the diameter of a circle to its circumference. Remember that the circumference equals Pie times the diameter? The length of an arc comprising a fraction of a circle is equal to the radius of the circle times the angle (in radians) of that slice of the pie that makes the arc. This is shown in Figure 1. below.

When we bring this geometric relationship to the shooting world, r is the distance to our target which would typically make the circle very big. We aren't concerned with the entire circle, but only the small section of arc length, s. The components of the circle above, are shown below in Figure 2 along with the addition of the bullet's flight trajectory and our calculated bullet drop.
Picture
Figure 2. Components of our shot on the circle.

In rifle shooting, the distance to target (r) and relatively flat trajectory of the bullet's flight makes the angular correction (Theta) small. As Theta becomes small, you can see that the arc length (s) becomes really close to the length of the straight line shown as the bullet drop. A somewhat complicated mathematical proof can show this, but we can also see it happening on Figure 2. You can see that if the angle Theta becomes a fraction of what is drawn, that the arc length and bullet drop components of the diagram would essentially overlap and be the same length for all practical purposes. This is known as the Small Angle Approximation and is the basis of how we make an angular adjustment to our optic to compensate for the linear correction of our calculated bullet drop. The numbers we work with in shooting are fractions of a degree, or thousandths of a radian. Small angles... Also, the relationships shown only work when we think and compute with the unitless angular measurement of radians. We can convert degrees to radians and proceed with the above math if need be.

The angular corrections our ballistic calculators and scopes talk in are Minutes of Angle and milradians. A minute of an angle is 1/60th of that angle. A second of angle would be 1/3600 of that angle. Talking in degrees, a MOA is 1/60th of a degree. A milradian is 1/1000th of a radian. In this case "mil" comes from the latin derived prefix of the metric system milli (and should technically be milliradians), not the military. There is a third and hopefully soon forgotten angular measurement called IPHY (inches per hundred yards) that still exists in some realms. IPHY isn't really an angular measurement and probably originally existed as an approximation of MOA that only works because our optics didn't adjust as well and our calculations were not as good. At any rate, I have not found a modern scope turret with these graduations so I think the industry agrees with me.

Back to the confusing lingo, we have two systems of angular measurements in use with many names for each. Minutes of Angle is often referred to in short form as "minutes" and "MOA". Milradians are commonly referred to as "Mils", "mildots", "mrad", and "milrads". Clicks would be each increment of graduation on a scope turret, of either system. Thinking in and communicating in clicks should be avoided to prevent confusion and error. Counting clicks to adjust for a correction when there are numbers on the turret is just silly. If one were to be confident in where their turret was currently at and could calculate the difference to their next come up, one could count clicks to get there in a case where they didn't want to take their eye off the target or be able to see the turrets in the dark.

Minutes of Angle and milradians are essentially two different languages to talk in for the rifle shooter. When I am asked which is better my answer is the question "What language are your friends speaking?" If you are shooting a Precision Rifle Series match with an MOA scope and everyone is talking wind holds in Mils, you are at a disadvantage unless you can quickly do the conversion in your head without mistakes. Vice versa at an F-Class match if you show up with a Mil scope. For the less established uses, either will work just fine. MOA scopes have finer increments of angular adjustment with 1/4 minute clicks than a Milrad scope with 1/10 Mil adjustments (like how I used every variant of the lingo there?). Even finer adjustment increments can be obtained with a scope set up with 1/8 MOA clicks which are typically used in Benchrest and F-Class competition, as well as varmint shooting. Using the Mil system lets the shooter remember a number like 8.7 mils rather than 29 7/8 MOA which can be a benefit in a high paced sport like the Precision Rifle Series.

We have discussed making these corrections with the scope's turret, but using a reticle with subtend marks can accomplish the same adjustment. You just need to know what the reticle graduations mean, and if it depends on the optical plane that the reticle is projected upon. With first focal plane scopes, the reticle changes with magnification and stays true to the objects in view. With second focal plane scopes, the reticle stays the same size to the eye and changes relative to the object in view as you change magnification. In the case of second focal plane scopes, you have to know what magnification the reticle subtends true at.

In today's market, it is hard to find a new scope that has mixed systems but thanks to the military being hard headed and a certain manufacturer being slow to adapt (cough, Leupold), those do exist. There are many scopes out there collecting dust with MOA turrets and Mil-Dot reticles. To use those scopes effectively, one had to truly understand the angular measurements and keep track of some conversions. Relating MOA to milrads is simple and starts with the following equation and breaking it down as follows:

360 degrees = 2 Pi radians
360 degrees (60 MOA/degree) = 2 Pi radians (1000 milrads/radian)
21600 MOA = 6283 milrads

1 MOA = .2908 milrad
or
3.438 MOA = 1 milrad

The other useful calculation is to see what distance our angular measurement covers on target. We will use the common sight in distance of 100 yards for this analysis. Making a jump from our above small angle approximation discussion, a milrad covers 1/1000th of the distance to the target. One radian would be the same as the distance to the target, except that is not a small angle anymore and the error of approximation becomes significant.

s = r Theta

in other words:

height on target per milrad = distance to target (1/1000)
height on target per milrad at 100 yd = 100 yds (1/1000)(36 in/yd)
height on target per milrad at 100 yd = 3.6 in

and to show this in the common turret graduation of 1/10 milrads:

height on target per 1/10th milrad at 100 yd = 3.6 in/10 = .36 in

The above math should quickly dissuade you of the myth that the milrad system is metric. It actually doesn't care what units of linear distance you use. The math is easier if we use the metric system but it works the same in any system. For example:

height on target per milrad at 1000 meters = 1000 meters (1/1000)
height on target per milrad at 1000 meters = 1 meter

or:

height on target per milrad at 16 fathoms = 16 fathoms (1/1000)
height on target per milrad at 16 fathoms = 16/1000 fathoms (6 ft/fathom) (12in/ft)
height on target per milrad at 16 fathoms = 1.152 in

If our units of distance to the target and height on the target are the same, we simply divide by 1000. It is when we start measuring in yards thinking in inches that requires the introduction of conversions.

We can use the MOA system to do the same math with the additional step of adding in the conversion of MOA to milrads. The commonly needed example would be the height on target at 100 yards that a MOA covers, which is found as follows, and starting from the work above:

height on target per MOA at 100 yards = 100 yards (1/1000)(36 in /yd)(.2908 milrad/MOA)
height on target per MOA at 100 yards = 1.046 in

The above relationship is the lure of the MOA system, where it just happens to work out that one MOA covers approximately one inch at 100 yards (and is where that IPHY concept came from in the past). Using this approximation:

1 MOA at 500 yards is approximately 5 inches
1 MOA at 1000 yards is approximately 10 inches and so on.

In summary, thanks to the small angle approximation we can use an angular dimension to move our point of impact or measure a linear distance on target. To move our point of impact, we just need to listen to our ballistics calculator and tell our scope how far to go. The measurement of a distance, whether it be a group size on target or the distance from the target that a miss strikes the dirt can be taken by using the scale in a scope's reticle and expressed as an angular measurement in Mils or MOA just the same. When working with a spotter, it is important to be thinking and communicating in the same language.

Now, why do they make scope bases marked 20 MOA?

Rifle scopes have an internal adjustment range. The adjustments can only go so far in either direction until the internal mechanism runs out of travel. Most scopes are made so that half of the adjustment range in in the "up" direction and half is in the "down" direction from their mechanical center but some examples exist where the engineers have designed them to be biased toward having more "up" than "down". A rifle setup doesn't need much "down" travel unless you were to try shooting while standing on your head. Some down is needed from the physical center of adjustment to get zeroed, but certainly not half of the entire range. Figure 3 shows a scopes internal adjustment range in an unbiased configuration.

For the following analysis, angles in the diagrams are greatly exaggerated.
Picture
Figure 3. Total Internal Adjustment Range of Scope

The line of sight (view of what you are aiming at) can be anywhere within the total adjustment range. Our long range math, and probably past experience with a football tells us that we have to point the barrel upward to start the projectile along its needed trajectory. Figure 4 shows the diagram from above with the bore pointed upward and the line of sight horizontal, the situation needed for a shot at distance. You can see that we use some of the "up" direction to accomplish this and at a point we would run out of scope travel to shoot further.

In the graphic above, you can also see how much of the total adjustment range is unusable. By canting the scope downhill relative to the bore axis, we can bias the total adjustment range to obtain more in the usauble, or "up" direction. This is shown in Figure 5 with the difference as some angle between the Bore Axis and Center of Adjustment Range. Scope bases are commonly available as 10, 15, 20, 30, and 40 MOA. For extreme distance shooting, bases can be custom made with much more cant built in but you can get into the case of not being able to sight in at a more conventional close range.

In the final diagram (Figure 6), the maximum "up" adjustment is shown by tilting the bore axis upward until the line of sight and maximum "up" lines coincide, yielding the furthest shot possible with turret adjustments with this rifle setup. You can see how the bore axis has gotten steeper to the line of sight in each step from Figure 3 - 6 allowing a higher trajectory.

Modern tactical rifle scopes built on 34 and 35mm tubes combine ample total adjustment range with the magnification needed for extended range shooting with common cartridges and high ballistic coefficient bullets. In extreme conditions, range can be gained by adding reticle holdover to turret adjustments. Most projectiles are falling fast in this realm, so gained distance can be minimal but it is entertaining to calculate a trajectory and attempt these shots for science. A second focal plane scope with the magnification backed off to half of the calibrated magnification doubles the amount of reticle holdover too...


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