# 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!

# Gun Basics for Writers: Modern Ammunition Basics, Part 2 of 2: Caliber and Gauge

I know I promised this post ages ago . . . sorry it took so long!

(Edited to add the pictures, which I forgot the first time.)

## Caliber

Okay, so.  Caliber!

From movies, we all hear things like “nine millimeter” and “a thirty-eight.”  What does it all mean?

In simplest terms, “caliber” refers to the width of the ammunition round.  Ammunition is commonly measured in both millimeters and decimal measures of inches, so a “nine millimeter” is nine millimeters in diameter, and a thirty-eight is (theoretically) .38 inches in diameter (and would usually be written “.38” and pronounced “thirty-eight”).

Sometimes, particularly for rifle ammo, there will be a second number.  For instance, 7.62×39 (“seven-six-two by thirty-nine”), the type of round fired by AK-47’s, is 7.62mm wide and 39mm long.  The “caliber” is only the 7.62 part, though—saying “7.62x39mm” is actually the cartridge name, since it specifies more than diameter (for example, 7.62x54mmR is a different cartridge from 7.62×39, but is also 7.62 caliber).

That’s the easy part of caliber.  Where most writers go wrong is not knowing how these relate to each other.

For instance, 9mm Luger, .38 Special, and .357 Magnum ammunition all have the same diameter (about .36 inches).  In general, ammunition of the same diameter is not interchangeable—there are other subtle differences, and guns are chambered for a specific type of ammunition—but the most common exception to this is that you can fire .38 Special out of a .357 (though not the other way around).

Why am I making a point of this?  Because I’ve seen TV show characters find a .357 revolver and say, “It can’t be the weapon; the murder was committed with a .38!” when it’s very common to fire .38 Special out of a .357.  I’ve also seen TV characters look at a bullet hole and know it’s a .38 instead of a nine-millimeter, which I find . . . unlikely, since the width of the bullet is pretty much exactly the same.

I’ve also seen people assume a wound from, say, a 9mm comes from a handgun.  Not true—for instance, MP-5’s fire 9mm.  So do Uzis.  And not even a different 9mm from most handguns!  (There are several types of 9mm, further complicating things if  you want to say “nine-millimeter” . . . for instance, the Makarov fires 9×18 ammunition, which is one millimeter shorter than 9mm Luger at 9×19.  But 9mm Luger is the most common variant of 9mm by far.)

And this is just the tip of the iceberg.  If you want to get into the nitty-gritty of all different ammunition types ever, it’s a mess.  Cartridges have evolved from each other over history and spread into a massive variety; different countries have come up with their own types of ammo; some cartridges have used casings inherited from others; old weapons end up re-chambered for modern ammo . . . it reminds me of language, the way cartridges have grown and crossed borders and evolved in a historical tree that’s almost impossible to follow.  I don’t know even a thimble-full of all the cartridge types in the world, and I do this for a living.

So for this post, I’m going to talk about some common ammunition types, common mistakes, and what types of weapons they’d be used in.  As always, this is the basics only, but most writers won’t need too much more.

# Weapon of the Day: ZOMBIE-KILLING CHAINSAW BAYONET.

One of the things I love about working in firearms is that I get to see and handle some insane weaponry.

Did I say insane?  I meant insane.  Because, hypothetically, haven’t you ever wanted a chainsaw attachment to mount on the underside of your assault rifle?  I THOUGHT YOU HAD.

The Zombie Chainsaw mounted on the underside of an M4. Image under copyright by Panacea X; I stole it from their website without permission. Since I’m advertising their product, I’m hoping they won’t come after me with a C&D letter. Or, you know, a chainsaw.  (Although I’m pretty sure this counts as fair use.  Critical commentary, yo!)

The company Panacea X is making chainsaws you can mount on the rails of your assault rifle.  Yes, these are real, working chainsaws, powered by battery packs.  Specifically marketed for the zombie apocalypse.

# Gun Basics for Writers: Modern Ammunition Basics, Part 1 of 2

I promised in my last gun basics post that I’d do a post on ammunition.  Here I’ll cover the very basics of vocabulary for modern ammo—there’s a lot more to know about ammunition, but this should get non-gun people started.

This post will be about the parts of an ammunition round.  Part 2 will cover the concept of caliber.

First, a little history.  Ammunition has gone from musket balls that were packed in a barrel with gun powder through an evolution that has led to modern cartridges.  Metal cartridges like those we use today started to be developed in the mid-nineteenth century and were adopted through the latter half of the 1800’s.

## Handgun and Rifle Cartridges

A modern round has four parts: the primer, the powder, the casing, and the bullet.  The casing is a cylindrical tube that holds everything together: the primer is in a tiny cap on the back of the cartridge and is what the firing pin drops onto, the powder goes inside the casing, and the bullet is at the other end of the casing from the primer, like so:

Ammunition cartridge diagram (Wikimedia Commons / public domain).
The firing pin hits the primer (5), which ignites the powder (3) which propels the bullet (1) out of the barrel. The casing (2) is left behind.
(The rim (4) generally allows for extraction of the casing; you wouldn’t need to worry about this in most fiction writing.)

When the gun is fired, the firing pin hits the primer. Continue reading

# Gun Basics for Writers: Types of Firearms – Basic Civilian Small Arms, Part 2 of 2: Long Guns

This is the third installment of my continuing series of gun basics for writers.  This is a continuation of the other week’s post in which I talk about the basic differences between weapons you’d be likely to find in civilian possession in the United States.  Again, these are the basics only—if you’d like more details on any part of this, feel free to request such in the comments! Last time we talked about the two broad categories of handgun.  Today we’ll talk long guns.  Long guns have two basic categories: shotguns and rifles.

# Weapon of the Day: The Duck Foot Pistol

Every once in a while I come across a weapon that’s so odd that I just have to stare at it and say, “No way.

The duck’s foot pistol is one of these.  I first encountered it a few years ago, and to this day it’s on the top of my So Odd It’s Fantastically Cool weapons list.  Why?  Well, here’s a picture:

Duck Foot Pistol. Credit: David Cushman’s website.

This gun knocks me out.

# Gun Basics for Writers: Types of Firearms – Basic Civilian Small Arms, Part 1 of 2: Handguns

Once again, this is written for the non-firearms person, and specifically for writers who may be totally confused about guns.  I’m going to start with the very basics.  If this is too beginner for you, feel free to request topics you’d really like to see!

To avoid this post becoming a book, I’m not going to cover military hardware or fully automatic weapons today—that will be another post.  Instead, I’m going to start with weapons that, in the United States, are generally permitted in civilian possession.  In California, for instance, these are the weapons that you can possess without obtaining a special license (minus the eyeroll-inducing Handgun Safety Certificate).

# Gun Basics for Writers: The Three Rules of Gun Safety

Being someone who works with firearms professionally, I’ve discovered just how little most people know about firearms.  So, here’s a new series for you: Gun Basics.  Because I’m coming at this blog in my persona as a writer and not in the context of the firearms community, I’m going to go a step further and say that this series is aimed at writers who are not even firearms owners (and may never intend to be firearms owners!) who want to get their gun scenes right.  I’m therefore going to talk a lot about how different firearms feel to shoot and what “sense”—what connotation, if you will—different weapons have.

But today, we shall start with the very first thing all shooters everywhere learn before they even pick up their first firearms: the rules of gun safety.