The fundamental definition of a radian: take a piece of string as long as the radius of a circle, and bend it round the circumference, The angle it subtends at the centre is a radian.
Because you need 2*pi radii to to go all round a circle, it follows that 2*pi radians = 360 degrees; 1 radian = 180/pi degrees.
An arc length L subtends an angle (L/r) radians where r is the radius of the circle.