Mils ↔ Degrees ↔ Grads Converter

Mils, Degrees, and Grads — Complete Angle Unit Guide

Every angle measurement system divides a full circle into a fixed number of equal parts. Degrees use 360, grads/gons use 400, and NATO mils use 6400. Each system was designed for a specific discipline — degrees for general use, grads for surveying arithmetic, and mils for military fire control. This converter handles all three instantly, along with radians and the Soviet 6000-mil standard.

What Are NATO Mils and Why Does the Military Use Them?

The NATO mil divides a full circle into 6400 equal parts. The number 6400 was chosen because it is highly composite — divisible by 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 64, 80, 100, 128, 160, 200, 320, 400, 640, 800, and 1600 — making arithmetic in the field fast and accurate. A right angle is exactly 1600 mils; half a right angle is exactly 800 mils; a back-bearing is always the original bearing plus or minus exactly 3200 mils.

The defining advantage of mils in artillery is the mil relation: at a range of 1000 metres, one mil subtends approximately one metre of width. At 500 metres, one mil subtends 0.5 metres; at 2000 metres, one mil subtends 2 metres. This linear relationship between mil angle and real-world width at known range is why fire-control systems, tank sights, rangefinders, and mortar aiming circles are all graduated in mils. A gun crew can instantly convert an angular correction into a lateral shift on target without trigonometry.

What Are Grads (Gons) and Why Do Surveyors Use Them?

The grad (also called the gon, from the French grade) divides a full circle into 400 equal parts, and a right angle into exactly 100 grads. This decimal relationship is the key advantage: in a closed traverse or building survey, angles at each station add up in a system directly compatible with decimal arithmetic and metric distances. European theodolites — particularly those from Leica (Wild), Zeiss, and Trimble's European product lines — are commonly graduated in grads.

France introduced the grad as part of the metric system reforms of the late 18th century. Today it remains the default angular unit in much of continental European surveying and is specified as the standard angular unit in several national cadastral systems. If your field book shows angles of 98.6412 g or your total station displays readings with a superscript g, you are working in grads. One grad equals exactly 0.9 degrees or 16 NATO mils.

How to Convert Between Mils, Degrees, and Grads

All conversions pass through the ratio of the two circle divisions. To convert degrees to NATO mils, multiply by 6400 ÷ 360 (≈ 17.778). To convert mils to degrees, multiply by 360 ÷ 6400 (= 0.05625). To convert grads to mils, multiply by 6400 ÷ 400 (= 16). The full set of exact conversion factors is:

• 1° = 17.77̄ NATO mils = 1.11̄ grad = π/180 radians
• 1 NATO mil = 0.05625° = 0.0625 grad = π/3200 radians
• 1 grad = 0.9° = 16 NATO mils = π/200 radians
• 1 radian = 180/π ≈ 57.2958° ≈ 1018.59 NATO mils ≈ 63.662 grad

Because 6400 and 400 share the common factor 400, and 6400 and 360 share the common factor 40, these fractions simplify to small exact rationals: 1 grad = 16 mils exactly; 1° = 160/9 mils exactly. This tool computes all conversions in 64-bit IEEE 754 floating point, displaying 6 significant figures for degrees and 4 decimal places for mils.

NATO Mils vs Soviet Mils — Why There Are Two Standards

The Soviet Union and Warsaw Pact countries adopted a different mil standard: 6000 mils per full circle rather than NATO's 6400. The motivation was to make 1 mil correspond more closely to a true milliradian (there are 2000π ≈ 6283.2 milliradians in a circle). The Soviet system accepts a slightly larger rounding error in the mil relation but produces mil values that are 6.25 % smaller than NATO mils for the same angle.

In practice the difference is significant: an azimuth of 1500 Soviet mils corresponds to exactly 90° (a right angle), whereas 1600 NATO mils equals 90°. Mixing the two systems in a fire-control calculation produces an angular error of about 6.25 %, which at 2000 m range represents a lateral miss of roughly 125 m — a critical blunder in combined arms operations. This converter displays Soviet mils as a reference output so you can cross-check values when working with legacy Eastern-bloc equipment.

Mils in Artillery: Fire-Control and Deflection

Artillery fire missions are expressed in azimuth (direction to the target) and elevation (firing angle). Both are measured in mils on the sight and aiming circle. A deflection correction — the lateral shift needed to move rounds onto a new target — is calculated in mils and applied directly to the gun's traversing mechanism. Because the mil relation holds at all practical ranges, a fire direction centre can compute deflection corrections from observer reports without knowing the exact range: add 1 mil of deflection for every metre of lateral correction at 1 km, 2 mils per metre at 500 m, and so on.

Modern computerised fire-control systems perform these calculations automatically, but the underlying mil arithmetic is preserved in training and in fallback manual procedures. Knowing how to convert 45° to 800 mils, or how to determine that a 15-mil correction at 3000 m moves a round 45 metres laterally, remains a fundamental field skill for artillery crews.