The_Derivation_of_the_Range_Estimation_Equations.pdf

MILS and MOA

A Guide to understanding what they are and

How to derive the Range Estimation Equations

By Robert J. Simeone

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 wondered. 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’s use

of mils, which 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.

We are also 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. Also, some new scopes, like mine, now 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 (Fig.1).

To find out how many “radians” (and /or radian arc’s) are in a circle, we use the

circumference formula of a circle, which is C = 2 r, where “r” is the radius. Take 2 r

and divide by r. (Note: = 3.14159)

2 r = 2 r = 2 = 2 x 3.14159 = 6.2832.

r r

Therefore, there are 6.2832 radians in a circle (or, 6.2832 radian arc’s that go

around the circumference of a circle) .

No matter how long the radius “r” is, there will always be 6.2832 radians in a circle

(because the “r” ’s always get cancelled out in the arithmetic and all you’re left with is

2 ).

How many degrees are in a radian (or how big is the angle created by the radian arc)?

Since there are 360 degrees in a circle and there are 6.2832 radians in a circle, then there

are: 360/6.2832= 57.3 degrees per radian in all circles no matter how long “r”is. (Fig. 2

below). Since there are 57.3°in each radian, and there are 6.2832 radians in a circle, then

6.2832 x 57.3° = 360° in a circle. Make sense?

Fig. 2

(A) Minutes in a Circle

There are 360 degrees in a circle, and each degree is composed of 60 minutes (60’).

Therefore, there are 360 (degrees) x 60 (minutes) = 21,600 minutes in a circle (21,600’).

(B) Milliradians (mils)

What is a milliradian? A “mil” is defined as “one thousandth”, or 1/1000. Therefore, a

millradian is 1/1000 of a radian . Take each of the radians that go around a circle and

chop it up into a thousand pieces. Since there are 6.2832 radians in a circle, and each

radian is chopped up into a thousand pieces, then there are 6.2832 x 1000 = 6,283.2

milliradians in a circle . (Milliradians is usually just shortened to “mils” )

I need to find out how many degrees are in each milliradian. A circle has 360 degrees,

and/or 6,283.2 milliradians that go around it ( B above). Therefore:

There are .0573 degrees per mil (degrees/mil)

(D) Minutes in a Milliradian (or minutes per mil)

I also need to find out how many minutes (referred to as minutes of angle, or moa) there

are in each mil. Let’s review. We have 21,600 minutes in a circle ( A above). And we

have 6,283.2 mils in a circle ( B above). Take 21,600 minutes and divide that by 6,283.2

mils and you will get 3.4377 minutes/mil. Let’s shorten that to just 3.438 minutes/mil .

Hang in there with me, just a few more things to figure out.

(E) Inches per Mil at 100 yards.

Look at the circle in Fig. 3 below. Make the radius 100 yards, or 3,600 inches long.

(1 yard = 36 inches. 100 yards x 36 inches = 3,600 inches in 100 yards).

Fig. 3

Remember earlier from Fig.1 (page 3) that all the sides of the piece of pie are equal.

Therefore, if one side is 3,600 inches, then all sides of the pie are also 3,600 inches (see

Fig. 4 below). So what is 1/1000 of any of those sides, which would also be 1/1000 of the

radian arc? Essentially, what is 1 mil equal to (remember, 1 mil is defined as 1/1000 of a

radian)? 3,600 inches/1000 = 3.6 inches. (The drawing is not to scale, but you get the

hint). Therefore, at 100 yards, 1 mil = 3.6 inches.

Fig. 4

For you math majors out there, another way to find the answer is to look at the bottom of

Fig. 4 as a triangle (enlarged in Fig. 5 below). We want to find the value of “x” in Fig. 5.

The way to do this is to use the tangent function of trigonometry.

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