MILS and MOA (Get comfortable this is long (15 pages) and some would say boring.) Page 1

A Mathematical Guide to Understanding Range Estimation Equations

The equations for determining the range to a target using mils, and with some new scopes, moa, are: Has anyone ever wondered how they came up with these? Well I’ve assembled them here in the paper. I’ve read numerous accounts on the internet on “mils”, but none of them explained how to actually derive these equations. They just seem to pull them out of the air at some point in their explanation of “mils” without actually showing us how or where they came from. I searched the internet far and wide, but to no avail. I was starting to wonder if they were derived so long ago that nobody knew how to do it anymore. So I decided to derive them on my own.

The reason I wanted to do that is because I feel that if you know how or why something works, or where it came from, you get a better understanding and appreciation of it, as well as its uses and limitations. In this paper I will attempt to explain, in simple English and math, how they came up with these equations. I will try to keep it, to the best of my ability, simple, methodical (painfully in some cases), slow and easy to understand. So here it is.

First, a brief history of mils: A “mil” is a unit of angular measurement. The military use of mils was used to help direct artillery fire, goes back as far as the late 1800’s. Its modern form of use by the military for directing fire was developed in the 1950’s. The modern “mil” is short for milliradian, a trigonometric unit of angular measurement. It is finer in measurement than degrees, thus more precise. In shooting, we can use mils to find the distance to a target, which we need to know, to adjust our shot. It is also used to adjust shots for winds and the movement of a target. (The actual techniques of how to use mils for shot adjustments are beyond the scope of this paper since this paper only deals with the math behind the equations). That’s the short….very short….history of why we have and use mils.

I’m going to talk about and define another term; minute of angle, or moa. It is another unit of angular measurement and used quite a bit in shooting. It is even smaller than a “mil”. Usually, we range our targets in mils and adjust our scopes in “minutes of angle”, or we talk about our “groupings” in moa. For instance, “my rifle shoots 1 moa all the time” or something like that. Most new scopes have reticles etched in minutes of angle (moa) rather than in mils. Therefore we will also derive the moa distance equation.

Before we can derive the equations, we need to define a few things and establish a few relations. So just follow along, hang in there, and you will see why we will need these later.

**Radians**

What is a radian? (Warning: you might have to read this paragraph a few times). A radian is a unit of angular measurement. Officially, one radian subtends an arc equal in length to the radius of the circle, “r”. (Yeah that helps). How about this. What a radian is it associates an arc length, called a radian arc, which is equal in length to the radius of the circle, with an angle at the center of the circle. The angle the arc created is called a radian. Or, another way, it’s the angle created at the center of a circle by an arc on the circumference of the circle, and that arc length is equal in length to the radius of the circle. Think of it as a piece of apple pie, where the two sides of the pie (the radii) are each equal in length to the curvature part of the pie (the arc). The angle created by the three sides at the center of the circle equals 1 radian. (Refer to figure 1 below)

The rest of the article is attached in PDF format. It contains several diagrams.