What is the difference between NATO mils, Soviet mils, and true milliradians?

All three are 'mils' but they divide the circle differently. NATO mil = 1/6400 of a circle (chosen because 6400 is easily divisible and maps cleanly to compass directions: N=0, E=1600, S=3200, W=4800). Soviet/Warsaw Pact mil = 1/6000 of a circle. True milliradian = 1/(2000π) ≈ 1/6283 of a circle — mathematically exact but less convenient for field calculations. NATO accepted a ~2% error at long range in exchange for easier mental arithmetic under stress.

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Complete Angle Unit Reference — All 15 Units Across 5 Categories

Unit	Symbol	Category	Divisions per Full Circle
Degree	°	Common / Mathematical	360
Radian	rad	Common / Mathematical	2π (≈ 6.283)
Turn / Revolution	rev	Common / Mathematical	1
Gradian / Gon	gon	Surveying / Metric	400
NATO Mil	mil (NATO)	Military / Artillery	6,400
Soviet / Warsaw Pact Mil	mil (SU)	Military / Artillery	6,000
True Milliradian	mrad	Military / Precision	2,000π (≈ 6,283.2)
Minute of Arc (MOA)	′ / arcmin / MOA	Precision / Astronomy / Ballistics	21,600
Second of Arc	″ / arcsec	Precision / Astronomy	1,296,000
Quadrant	—	Navigation / Classical	4
Sextant	—	Navigation / Classical	6
Octant	—	Navigation / Classical	8
Compass Point	pt	Navigation	32
Hour Angle	h	Astronomy	24
Binary Degree	brad	Computing / Robotics	256

NATO 6400 vs Soviet 6000 vs True Milliradian — Why There Are Three "Mils"

A milliradian (mrad) is mathematically defined as 1/1000 of a radian. Since a full circle = 2π radians, a full circle = 2π × 1000 ≈ 6,283.185 milliradians. This is the mathematically exact value — but it's terrible for battlefield mental arithmetic. NATO chose 6,400 mils per circle instead — a deliberate fudge. Why 6,400? Because it makes compass navigation trivially easy: North = 0 mils, East = 1,600, South = 3,200, West = 4,800. The 6,400-division system maps cleanly onto the four cardinal directions (multiples of 1,600) and subdivides easily by 2, 4, 8, and 16 — exactly what a soldier needs when calculating firing solutions under stress. The cost? A 1.86% systematic error: at 1,000 meters, a NATO mil subtends 0.9817 meters instead of 1.0000 meters. The Soviet Union chose 6,000 mils per circle — simpler for decimal-oriented Warsaw Pact armies but incompatible with NATO equipment and training. This converter shows all three side by side, so you can see exactly how much the numbers diverge at any given angle.

MOA (Minute of Angle) — The Shooter's Ruler

1 MOA = 1/60 of a degree = 1/21,600 of a full circle. At exactly 100 yards, 1 MOA subtends 1.047 inches — close enough to "1 inch at 100 yards" that the shooting community treats this approximation as a practical truth. A rifle that shoots "sub-MOA" means its group size at 100 yards is under 1.047 inches. Because MOA is an angular measurement, it scales linearly with distance: 1 MOA ≈ 2.09″ at 200 yards, ≈ 5.24″ at 500 yards, ≈ 10.47″ at 1,000 yards. This makes MOA a universal language for precision shooting — a 2 MOA adjustment on a scope means the same angular change regardless of the target distance. The reference table below translates MOA into inches at common shooting distances:

MOA	at 100 yd	at 200 yd	at 300 yd	at 500 yd	at 1000 yd
0.25 (1 click)	0.26″	0.52″	0.79″	1.31″	2.62″
0.5	0.52″	1.05″	1.57″	2.62″	5.24″
1	1.05″	2.09″	3.14″	5.24″	10.47″
2	2.09″	4.19″	6.28″	10.47″	20.94″
5	5.24″	10.47″	15.71″	26.18″	52.36″
10	10.47″	20.94″	31.42″	52.36″	104.72″
20	20.94″	41.89″	62.83″	104.72″	209.44″

Formula: MOA × (distance in yards / 100) × 1.047 = spread in inches. At 100 meters, replace yards with meters and multiply by 1.145 instead of 1.047.

Gradians — The French Revolution's Failed Decimal Angle

In the 1790s, alongside the metric system for length and weight, French revolutionaries attempted to decimalize angles. The gradian (or gon) divides a right angle into 100 units and a full circle into 400. This makes angle arithmetic beautifully decimal: 50 gon = 45°, 100 gon = 90°, 200 gon = 180°. Every surveying calculation becomes as simple as metric length calculations. But 400 has far fewer divisors than 360 — you can't cleanly express 1/3 of a right angle (30°) in gradians (33.333… gon). This irritates mathematicians, educators, and anyone doing geometric construction. The degree survived because 360 is a highly composite number — divisible by 2,3,4,5,6,8,9,10,12,15,18,20,24,30,36,40,45,60,72,90,120,180 — making mental subdivision effortless. Gradians retreated to a single surviving niche: European land surveying equipment. If you pick up a Leica or Trimble total station in France, Germany, or Switzerland, it defaults to gon mode. Surveying students in Europe still learn to think in gradians, even as the rest of the world uses degrees.

Why 360 Degrees? The Sumerian Legacy

The 360° circle traces back to the Sumerians and Babylonians (~2400 BCE), who counted in base-60 (sexagesimal). They noticed the year was roughly 360 days, and the sun appeared to move about 1/360 of the sky each day. More importantly, 360 is extraordinarily factorable — you can divide a circle into halves, thirds, quarters, fifths, sixths, eighths, and dozens of other clean fractions without ever needing a decimal point. This practical advantage is why 360° has survived every attempt to replace it — from the French gradian (400) to the radian (2π) to the binary degree (256). Each alternative has its domain of superiority, but none matches the degree's combination of intuitive size (1° is about the width of your little finger at arm's length) and arithmetic convenience.

Common Angle Conversions at a Glance

From	To	Multiply By	Example
Degrees	Radians	× π / 180 (≈ 0.017453)	180° = π rad
Radians	Degrees	× 180 / π (≈ 57.2958)	1 rad = 57.30°
Degrees	Gradians	× 10 / 9 (≈ 1.11111)	90° = 100 gon
Degrees	NATO Mils	× 6400 / 360 (≈ 17.7778)	90° = 1600 mil
Degrees	MOA	× 60	1° = 60 MOA
NATO Mils	Soviet Mils	× 6000 / 6400 (≈ 0.9375)	1600 NATO = 1500 Soviet
NATO Mils	True mrad	× 2000π / 6400 (≈ 0.9817)	6400 NATO ≈ 6283 mrad
Gradians	Degrees	× 0.9	100 gon = 90°
Compass Points	Degrees	× 11.25	1 pt = 11.25°
Hour Angle	Degrees	× 15	1 h = 15°
Binary Degrees	Degrees	× 360 / 256 (≈ 1.40625)	256 brad = 360°
MOA	Inches @ 100 yd	× 1.047	1 MOA = 1.047″

Frequently Asked Questions

How do I convert degrees to radians?

Enter your value in the Degree (°) field. The Radian (rad) field updates instantly. Formula: degrees × π / 180 = radians. Key values: 180° = π rad (3.14159), 90° = π/2 rad (1.5708), 45° = π/4 rad (0.7854). For rough estimates, divide degrees by 57.3.

What is the difference between NATO mils (6400), Soviet mils (6000), and true milliradians?

All three measure angles in "mils" but use different circle divisions. NATO mil = 1/6400 circle — chosen for clean compass mapping (N=0, E=1600, S=3200, W=4800). Soviet mil = 1/6000 circle — simpler decimal arithmetic for Warsaw Pact forces. True milliradian = 1/6283 circle — mathematically exact (2π×1000) but impractical for mental math. NATO accepts a ~2% error for battlefield simplicity. This converter shows all three simultaneously.

What is MOA (Minute of Angle) and why do shooters use it?

1 MOA = 1/60 degree = 1/21,600 of a circle. At 100 yards, 1 MOA ≈ 1.047 inches — close enough to 1 inch that shooters use it as a practical rule of thumb. Scope adjustments are typically in 1/4 MOA clicks (0.26″ at 100 yards). MOA scales linearly: 1 MOA = 2.1″ at 200 yd, 5.2″ at 500 yd, 10.5″ at 1000 yd. See the MOA distance reference table above.

What are gradians/gons and where are they still used?

1 gradian (gon) = 1/400 circle = 0.9°. Introduced during the French Revolution (1790s) as a decimal alternative to degrees: 100 gon = right angle. They largely failed outside of European land surveying — Leica and Trimble total stations default to gon mode in continental Europe. Most European surveying students learn to work in gradians. The reason 360° won: it has far more divisors (2,3,4,5,6,8,9,10,12,15,18,20,24,30,36,40,45,60,72,90,120,180) than 400 (2,4,5,8,10,16,20,25,40,50,80,100,200).

Why are there 360 degrees in a circle?

The 360-degree circle originates with the Sumerians and Babylonians (~2400 BCE), who used base-60 mathematics. They chose 360 because: (1) it's close to the 365-day year, (2) the sun appears to move ~1° per day across the sky, and (3) 360 is a highly composite number — it can be evenly divided by 24 different integers, making mental geometric subdivision dramatically easier than with any alternative system.