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Understanding Your Rifle Scope

Milliradians (mils) and Minutes of Angle (MOA)

Radians and degrees are both angular representations of height / distance and are therefore useful to shooters. Both units of measurement are useful. Milliradians are thousandths of radians (radian / 1000). Radians are most often used in engineering and by the military for artillery calculations. Minutes of angle are 1/60th of a degree. Degrees are most often used in very basic math and map reading since the magnetic compass uses degrees. A short review of some basic math follows, plus a hint at it's usefulness for shooters.

Note: The artillery mil is 1/6400th of a circle = 3'22.5" of arc, or 3.375 minutes. - ref: Hatcher's Notebook

What are radians?

C = 2 Π r The circumference of a circle is equal to 2 * Π * the radius of the circle. Therefore, to find the number of radians in a circle we solve:
    (2Πr)/r = (2Πr)/r = 2Π = 2 * 3.14159 = 6.2832 radians in a circle.
A milliradian is a thousanth of a radian so:
    6.2832 * 1000 = 6283.2 milliradians in a circle.

Why is this useful?

tangent = opposite / adjacent and at small angles tangent = radians.
Therefore, to find the distance to a target (range estimation) we solve:
    distance = height of target / tangent = (height of target / mils) * 1000.

Remember this:

Since opposite = tan * adjacent = 0.001 * 100 * 36 = 3.6
    1 mil = 3.6 inchs at 100 yards = 36 inchs at 1000 yards.

What are minutes of angle?

There are 360 degrees around a circle and 60 inches/degree.
Since there are 360 degrees in a circle:
    360 / 6283.2 = 0.0573 degrees/mil.
Since there are 60 minutes in a degree:
    (360 * 60) / 6283.2 = 3.438 minutes of arc/mil.

Remember this:

Since 3.6 inch/mil / 3.438 minutes/mil = 3.6 / 3.438 = 1.047 inch/minute:
    1 moa ≈ 1 inch at 100 yards ≈ 10.5 inches at 1000 yards.

Why is this useful?

Most rifle scopes adjust in 1/4 inch or 1/4 MOA increments. The angular measurement characteristics of the MOA translates nicely for range compensation. 1 inch at 100 yards is 2.5 inches at 250 yards, etc.

Remember this:

Since a 36 inch target at 1000 yards = 1 mil and a 36 inch target at 500 yards = 2 mils a rule of thumb is
    Double the distance and half the angle (mils or moa).
    Half the distance and double the angle (mils or moa).

The Effects of Magnification

2X magnification makes an object appear twice as large, effectively making it appear to be at half the distance. 4X doubles the 2X magnification and effectively making it appear to be at half the apparent distance or 1/4 the real distance.

Remember this:

Since 2X magnification makes an object appear twice as large, or appear to be at half the distance
    Double the magnification and double the angle (mils or moa).
    Half the magnification and half the angle (mils or moa).

Rifle Scope Reticles

There are quite a number of scope reticle designs. The mil dot scope has dots spaced along the cross hairs. Others may have fewer marks but like the one shown to the left, those marks may be significant in that they are a reference for angular measurement.

The manufacturer may tell you something like the distance between the dots equals one mil or the size of a dot is 1/8 mil, or the distance between the junction of the thick and thin part of a cross hair and the centre intersection of cross hairs is one mil. If it is a variable power scope that will be qualified with range. If you don't have the book or the manufacturer did not state in the book what those references are, there is a way to figure it out.

To Make a Reference Target

Choose a convenient distance for your trials and then using a calliper, measure the distance for a square the size of a calculated mil for that distance (use the calculator to the right). Fill the square so that the target will be high contrast to the background. If you want to see if there is a 1 MOA reference (cross hair perhaps), make a cross with line equal to the calculated MOA.

Using The Reference Target

While looking at the target through the scope at the chosen distance, match up the target size with a reticle reference. If you have a variable power scope, adjust the magnification to obtain a match and then note the magnification setting (6X for example).

Calculators

Google Pages doesn't seem to allow embedding javascript, arrggg! I'll gave to figure out Google Gadgets to make these calculators work.

Inches for 1 mil at [  ] feet. [calculate]
Inches for 1 moa at [  ] feet [calculate]

For a 0.5 mil square, half the distance.
For a 2 mil square double the distance.

With the target shown on the left [enlarge] and a scope with the reticle shown above I was able to determine that (1) the thick part of the cross hair = 1 MOA at 6X and (2) the distance from the intersection of the cross hairs to the point where the cross hair changes from thin to thick = 2 mils at 6X.

It is difficult to obtain a good focus with the scope that I used (3x9x40) at magnifications above 3X at 25 feet. With that target and higher minimum magnifications it might be necessary to double the target distance to 50 feet and half the reference size. The target marked 2 MOA becomes 1 MOA and the target marked 2 mil becomes 1 mil, etc.

The following chart can be constructed from the information obtained from these trials and some simple math using the information that has been presented. Most of the numbers are not memorable so it might be wise to make a chart that can be attached to the scope or rifle stock.

Fig. 1: mils & reticle marks by magnification
  3x 3.5x 4x 4.5x 5x 5.5x 6x 6.5x 7x 7.5x 8x 8.5x 9x
MOA Reference 2 1.71 1.5 1.33 1.20 1.09 1 0.92 0.86 0.80 0.75 0.71 0.67
Mil Reference 4 3.43 3 2.67 2.40 2.18 2 1.85 1.71 1.60 1.50 1.41 1.33

Range Calculation

Since tangent = opposite / adjacent we can transpose the formula so that adjacent = opposite / tangent. As pointed our earlier, at small angles the tangent equals radians so with correction multipliers to adjust for units of measure we can get the following range formulas. It is a happy coincidence that 1 mil at 1000 yards = 36 inches and i yard is 36 inches. So to that 1 MOA at 100 yards ≈ 1 inch. Or is it a coincidence?

Remember this:

Since a 36 inch target at 1000 yards = 1 mil:
    Distance (yards) = (Target height (yards) / mils) * 1000

Since 1000 / 36 = 27.78 the formula changes to something that would be difficult for most people to solve in their head in the field.
    Distance (yards) = (Target height (inches) / mils) * 27.78 &larr might not need to memorise this one

Since there are 3.438 minutes of arc/mil and 1.047 inch/minute, or D = ((inch * 27.78 * 3.438) / MOA) * / 1.047 the formula becomes
    Distance (yards) = (Target height (inches) / MOA) * 100 ← MOA reference and high magnification to be useful.

Example applications:

A good sized buck ≈ 18 inches back to chest near the shoulders is 1/2 of the mil reference (thick meets thin point on cross hair to the point where cross hairs intersect) at 6X. → target is 0.5 yards height and measures 1 mil (1/2 the reference 2 mils at 6X) → 0.5 / 1 * 1000 ≈ 500 yards range.

A coyote ≈ 12 inches back to chest near the shoulders is equal to the mils reference at 7.5X. → target is 0.33 yards height and measures 1.6 mils (reference 1.6 mils at 7.5X) → 0.33 / 1.6 * 1000 ≈ 206 yards range.

You've worked up a new load and want to sight in at 100 yards and check group size out in the field instead of going to a range. Your target is a 1 inch square. You pace off 100 one yard steps (the terrain is irregular) and then take a sight picture. You can just cover your target square with the thick part of the cross hair (MOA reference) when you adjust to 6.5X. → target is 1 inch height and measures 0.92 MOA (the reference 0.92 MOA at 6.5X) → 1 / 0.92 * 100 ≈ 108.7 yards. → You walked too far. Move back toward the target and test for 1 MOA reference at 6X.

Holdover, Windage, and Lead Adjustments

To use this information you must get data for your rifle and load. You can download a calculator here that is written in Python which means that you can use it on most computers and operating systems. You will need to download Python if you don't already have it. It' free and useful for many things. The Python script outputs a csv file for you to read with a spread sheet for formatting, selecting the data that you want (the script tells you more than you want to know). If you don't have a spread sheet program you can get an entire office suite at Open Office. It's also free and runs on many platforms. I like free, I use linux as an OS. It's free too! You can also use one of the many web based calculators (search engines are your friend) if you wish and then use the tools on this page to get mil numbers.

With the data obtained from the drop tables, and the information found here, you can make a table like the one in fig. 3 that follows. This table uses distances that go out to the limits of ethical hunting with this load. The rifle, scope, and load are clearly for large game at medium distances.

You can make a chart like this, or one that goes out to fantasy distances, as you wish. Calculate drop and drift by dividing drop or drift in inches by the size of a mil at that range from fig. 1. For example, at 100 yds the bullet strike is 3.45 inches high and one mil is 3.6 inch so 3.45 / 3.6 = 0.96 (your hold over is about 1 mil).

To add the column for leading a moving target you use the time of flight (TOF) for that range along with the number of inches per mil from fig 1. 1 MPH = 1.466667 ft/sec = 17.6 in/sec. So the formula is lead in mil per MPH = TOF / (17.6 / inch per mil). For example at 350 yds the TOF is 0.45 sec and there are 12.6 inches per mil so lead in mils = 0.45 / (17.6 / 12.6) = 0.45 / 1.4 = 0.32 mils.

Google Pages doesn't seem to allow embedding javascript, arrggg!
I'll gave to figure out Google Gadgets to make this calculator work.

To calculate the value of mils for [inches] at [yards], enter values and [click here]

Fig. 2: Range, Drift and Lead Compensation
RangeYards Impact
Inch
Impact
mils
Drift
10 mph
Inch
Drift
10 mph
mils
TOF
sec.
Lead
1 mph
mils
0 -1.50 NA 0 0 0 0
25 0.23 0.255 0.47 0.52 0.03 0.002
50 1.65 0.92 0.55 0.33 0.06 0.01
100 3.45 0.96 0.92 0.26 0.12 0.02
150 3.81 0.64 1.54 0.29 0.18 0.06
200 2.70 0.38 2.41 0.33 0.25 0.10
250 0 0 3.55 0.39 0.31 0.16
300 -4.34 -0.40 4.96 0.46 0.38 0.23
350 -10.41 -0.83 6.66 0.53 0.45 0.32
400 -18.30 -1.27 8.65 0.60 0.52 0.43
450 -28.10 -1.73 10.95 0.68 0.59 0.54
500 -39.92 -2.22 13.56 0.75 0.66 0.68
550 -53.85 -2.72 16.51 0.83 0.74 0.83
600 -70.02 -3.24 19.79 0.92 0.82 1.01
650 -88.54 -3.78 23.44 1.00 0.90 1.20

Laser Bore Sighting

Once you've made the chart shown above, you can use it to bore site your rifle. Bore sight instructions for a rifle with a scope height of 1.5 inch normally tell you to bore sight the cross hairs onto the laser dot (or bullet strike if sighting in at a range without a laser bore sight tool) as an approximate 200 yard sight in. For the load shown above the calculations from the link provided this would be sighting in for 216 yards. To sight the scope for 250 yards you would want the bullet strike at 25 yards to be 0.23 inches high or 0.255 mils. At 6 X the scope in this example the laser dot would be 1/16 of the distance from horizontal cross hair to the thick part of the vertical cross hair. Basically, the laser dot would sit on top of the horizontal cross hair. This may seem trivial but if you wish to sight this load in for 500 yards you would need the laser dot to appear 2.5 mils above the horizontal cross hair. That is a 45 inch difference at 500 yards.

Practical Considerations

We all know that the rock steady scope views with the target right under the cross-hairs in movies are total BS. In real life our target won't be exactly at the range that we sighted our rifles at, the wind may be blowing and estimating it's speed is difficult. And of course, we have wobble. At 9x, even off a sandbag rest I can see my pulse in the cross-hairs, and it's distracting. The table at fig. 3 has far more accuracy than we can hold, estimate the fractions of a mil, or accurately estimate the true size of the target, all of which contribute to error.

Having said that, the mil dot system has great advantage for those who know how to use it. Managing holdover, lead, and drift in mils is easier than trying to do it in inches. For the rifle and scope shown above, sighted in at the range shown, mil adjustments in aim are within a manageable value within the expectations of the table.

For a 400 yard shot the holdover would be approximately 1.25 mils and a 8 mph crosswind would call for a about 0.5 mil adjustment. That doesn't look like much, but at that range it's an 18 inch elevation correction and a 7.2 inch windage correction. Less than perfect, but better than otherwise under hunting conditions and in this case, the difference in a miss and a hit.

If you want to know about estimating wind speed or target speed, search engines are your friend. This article is focused on how you can use a standard variable power scope to do things that are normally reserved for more specialised, and expensive scopes. Money is tight, learn to use what you've got.

So there we have the mil dot system with a non-mil dot scope, or a mil dot scope for that matter. The mil system is based upon angles so it is attractive to shooters because it provides a consistent scope picture reference for range, drop, wind, and lead computation that is practical in the field. Happy shooting.

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