Mathematical principle Milliradian

for small observed angles (green) arc length (blue) approaches subtension (orange).

use of milliradian practical because concerned small angles, , when using radians small angle approximation shows angle approximates sine of angle,



sin
⁡
θ
≃
θ


{\displaystyle \sin \theta \simeq \theta }

. allows user dispense trigonometry , use simple ratios determine size , distance high accuracy rifle , short distance artillery calculations using handy property of subtension: 1 mil approximately subtends 1 meter @ distance of 1 thousand meters.

more in detail, because




subtension

≃

arc length



{\displaystyle {\text{subtension}}\simeq {\text{arc length}}}

, instead of finding angular distance denoted θ (greek letter theta) using tangent function

θ

trig


=
arctan
⁡


subtension
range




{\displaystyle \theta _{\text{trig}}=\arctan {\frac {\text{subtension}}{\text{range}}}}

,

one can instead make approximation using definition of radian , simplified formula:

θ

rad


=


subtension
range




{\displaystyle \theta _{\text{rad}}={\frac {\text{subtension}}{\text{range}}}}

since radian mathematically defined angle formed when length of circular arc equals radius of circle, milliradian, angle formed when length of circular arc equals 1/1000 of radius of circle. radian, milliradian dimensionless, unlike radian same unit must used radius , arc length, milliradian needs have ratio between units subtension thousandth of radius when using simplified formula.

approximation error

the approximation error using simplified linear formula increase angle increases. example, a

3.3 × 10 % (or 6993330000000000000♠0.00000033%) error angle of 0.1 mil, instance assuming 0.1 mil equals 1 cm @ 100 m
0.03% error 30 mils, i.e. assuming 30 mils equals 30 m @ 1000 m
2.9% error 300 mils, i.e. assuming 300 mils equals 300 m @ 1000 m

the approximation using mils more precise using common system 1′ approximated 1 inch @ 100 yards, comparably there a:

4.5% error assuming angle of 1 equals 1 in @ 100 yd
55% error 100 , i.e. assuming 100 equals 100 in @ 100 yd
953% error 1000 , i.e. assuming 1000 equals 1000 in @ 100 yd