Guns and Math: Does 1 MOA *Really* Equal 1 Inch at 100 Yards?

I’m doing rifle marksmanship training right now, and the rule of thumb for sight adjustment is that 1 minute of angle (MOA) equals one inch at 100 yards.  That means that if you adjust your sights to sweep in a certain direction (left, right, up, or down) by an angle measuring one minute, your point of impact on a target 100 yards away will move in that direction by 1 inch.  So, for example, if you adjust your rifle by 1 minute of arc left, your point of impact will shift left by 1 inch on the 100-yard target.

Of course, being a mathematician, my thought is automatically, “One MOA equals 1 inch at 100 yards?  How convenient!  I should do the math to see how close it actually is.”

It’s pretty easy to convert the angle measure of an arc to the arc length.  Take a look:

Here, theta is the angle, r is the radius (imagine the pie slice as a sector of a circle with radius r), and s is the length of the arc. (Wikimedia Commons, public domain.)

If you have the measure of the central angle in radians, the measure of the arc length — denoted s here — will be that number times the radius:

$s = \theta r$

Since the angles we’re looking at are so small, we can use a small angle approximation to say that our length of shift on the target equals the arc length —

. . . OKAY FINE, I shudder when physicists use the small angle approximation, but when I did it ABSOLUTELY EXACTLY without using the small angle approximation, the length of the chord matched the length of the arc out to 7 decimal places, which means our error here is about the width of an atom.  SO FINE.  We’ll use it and say the chord approximately equals the arc.

The red curve is the “s” we’re finding; the blue line is the actual straight-line distance on the target. For angles as small as we’re examining, the difference in length between them is about the width of an atom or two.

Let’s go back to the above formula.  We need one minute in radians.  One minute is 1/60 of a degree, so we get

$\left(1/60 \text{ degrees}\right) \left(\frac{\pi}{180 \text{ degrees}}\right) = 0.000290888209 \text{ radians}$

$s = 0.000290888209 * \left(100 \text{ yards}\right)$

$s = 0.0290888209 \text{ yards}$

(Look at me, truncating before we get out to 7 decimal places so the small angle approximation holds!  To find the exact number, you’d have to do 2 times the sine of half the angle you’re looking at (so, 2 times the sine of half of 1 minute converted to radians, in this case), and then multiply by the radius (100 yards) and then convert to inches.  Again, for angles this small it’s equal to the above out to an absurd number of decimal places.)

So, yeah, 1 MOA is 1.0472 inches of distance at 100 yards — it’s pretty close to one inch!

The rifle marksmanship rule of thumb continues to say that we’re at 1 inch of point-of-impact change per 100 yards out, so at 200 yards 1 MOA would be equivalent to 2 inches on the target, at 300 yards 1 MOA would be equivalent to 3 inches on the target, etc. (and at 50 yards or 25 yards, 1 MOA would be equivalent to .5 inches or .25 inches on the target respectively).  Here’s what the actual numbers are:

Distance 1 MOA Equivalence (Rule of Thumb) 1 MOA Equivalence (True)
25 yards .25 inches  0.2618 inches
50 yards .5 inches  0.5236 inches
100 yards 1 inch  1.0472 inches
200 yards 2 inches  2.0944 inches
300 yards 3 inches  3.1416 inches
400 yards 4 inches  4.1888 inches
500 yards 5 inches  5.2360 inches
1000 yards 10 inches  10.4720 inches

So we’re edging up to half an inch difference at 1000 yards.  That’s a lot!

Now the question becomes — well, scopes often adjust as 1 click = 1/4 inch.  But are they adjusting 1/4 of an MOA (and thus 1/4 of 1.0472 inches), or 1/4 of a true inch?  Wikipedia had the answer:

One thing to be aware of is that some scopes, including some higher-end models, are calibrated such that an adjustment of 1 MOA corresponds to exactly 1 inch, rather than 1.047″. This is commonly known as the Shooter’s MOA (SMOA) or Inches Per Hundred Yards (IPHY). While the difference between one true MOA and one SMOA is less than half of an inch even at 1000 yards,[5] this error compounds significantly on longer range shots that may require adjustment upwards of 20-30 MOA to compensate for the bullet drop. If a shot requires an adjustment of 20 MOA or more, the difference between true MOA and SMOA will add up to 1 inch or more. In competitive target shooting, this might mean the difference between a hit and a miss.

Hey, look how useful math is!