This very useful relationship follows directly from
the equation for the circumference of a circle and from the definition
of a radian.
Circumference = 2 times PI times the radius
Circumference is a special case of arc length. It
is the arc length if we go all the way around a circle.
If we go half way around a circle, arc length equals
half the circumference. If we go a quarter way around, arc length
equals circumference divided by four. And so on…
A radian is a unit of angular measurement such that
2xPI of them equals a full circle.
Therefore:
Arc length = angle in radians times the radius
The small angle equation uses the angle in arc seconds.
In order to use the arc length equation, we must convert arc seconds
to radians. The number of arc seconds in one radian is:
(360 degrees per 2PI radians)(60 min per degree)(60
sec per min) = 206264.8062 arc sec per radian
We usually round this off to 206265 arc sec per radian.
If we call the arc length d, the radius D, and the angle in arc
seconds alpha:
d = (alpha)(D)/206265
This is the small angle equation.