Section Leaders:
Fionn Kelly & Madmatt

Measuring Angles - The Army Way

By: Jon Sowden

Most modern armies use a variation of the mils system to measure angles. Artillery Forward Observers in particular use a sound knowledge of mils to get accurate target grids, and to speed up the adjustment procedure. But, what is the mils system, and why is it used?

To understand why mils are used, we need to consider the other main method of measuring angles – degrees.

Degrees

Most civilians and sailors, when they think about angles, think in terms of degrees, and as we all know there are 360 degrees in a full circle. To measure compass directions, North is usually taken as being at 360° (or 0°), South is 180°, South-West is 235°, etc. In most cases 1° provides fine enough steps to accurately measuring the required angle – especially on the heaving deck of a ship! Sometimes, however, greater accuracy is required in which case finer divisions called ‘minutes’ and ‘seconds’ are used. If anyone knows what size angle these measure, and the relationship between them, please let me know since I don’t have a clue! I suspect the same holds for most of the population.

If you remember back to your school days, you might recall something called SOH/CAH/TOA – a way to figure out the lengths of sides on a triangle, or the size of internal angles. In the military world this has enormous usefulness. It is generally fairly easy to figure out two of the three unknowns, and use those to find the third. For example, using binoculars with accurate markings it is easy to find the angle between two distant objects, and using the known size of the object under observation the distance between those points can be determined.

Then, using TOA … 

TAN(observed angle) = (known distance between points) / (distance to object)

… rearranging to give …

(distance to object) = (known distance between points) / TAN(observed angle)

… and inserting the values we can get the distance to the object.

To give a more specific example, consider a stationary train, viewed side on. We know, from previous research, that the train is 150m long. When looking through our binos the observed angle between the front of the engine and the end of the guards van is about 4.3°.

Using our formula:

(distance to train)   = (150) / TAN(4.3)

                                    = (150) / 0.07519

                                    = 1995m

Piece of cake. With a bit of thought it can be seen that the same principles you can figure out not only how far away something is, but how big it is, how fast its going, how tall it is, etc. The only proviso is that you must know at least two pieces of information about the object to start with.

As it stands, this process would be wonderful - except that for most soldiers this is nearly useless. Firstly, soldiers would need to carry a calculator around in the field, which is never going to be a winner – even assuming it doesn’t get broken it isn’t always practical to whip out the old Casio Fx-82 and start furiously calculating. Secondly, since there are only 360° in a circle, the accuracy with which angles can be measured isn’t that great unless minutes and seconds are also used - yeah, right!

Mils

This is where the mils system comes in. Under this system, the unit of angle – the mil – has been chosen such that there is no need to mess around with Sines, Cones and Tangents. Originally it was developed by the German military, was used by them during World War Two, and has since found its way into most armies.

The cornerstone in making the mil useful was in the choice of angle. 1mil is the angle formed by a right angle triangle 1000 metres long and 1 metre wide. Actually, the exact angle for a mil is a little different – using precisely 1:1000 leads to there being 6283.2 mils in a full circle. This is impractical for field use, so the number actually used is a rounding: in the Whermacht, and subsequently in NATO armies, this number was rounded to 6400. In Warsaw Pact armies the number for a full circle is 6000. Returning briefly to degrees, 1° » 18mils (NATO style).

Using 1:1000 makes the mathematics needed really simple. Going back to our example of the train, we know its 150m long, and looking through our binos calibrated for mils the angle between the front and the back is 75mils. Now, if the train were 1000m away we would expect to see it as being 150mils wide. Our figure is exactly half of that so the train is twice that distance away – 2000m. This is the same figure as before (give or take 3%), but is much easier to arrive at, and requires no special equipment other than a knowledge of how mils work.

How do I measure angles in mils?

Virtually every piece of kit the army has that is used to assist observation – binoculars, scopes, gunsights, and so on – is marked with graticles spaced in mils. This is what those funny looking ‘scissor’ binoculars you sometimes see next to German artillery in WW2 photos are. Using such equipment it is very easy to accurately measure the angle between points. However, even the regular grunt in his foxhole has a piece of kit calibrated in mils – his hand.

Figure 1: Hand Angle for a Hand held at Full Armstretch

By holding your hand out at full stretch and using the distance between fixed points you can readily get a fairly accurate idea of the angle between two points.

If you play golf, you might find it useful to get a feeling for the vertical hand angle of the pin at various distances. It makes selecting the right club much easier!