Radian Measure
Formally, we define a degree as 
of a straight angle. We use the symbol 
to denote degree measures: 
. However, in real-world applications, we often use angle measure to compute distances. We’d like to measure angles with a comparable scale (ie, with the same units as the other objects).
Thus we say a central angle of a circle has a measure of 1 radian if it intercepts an arc with the same length as the radius of the circle (see picture):
Radians and the Unit Circle
We’re often interested in working with circles of radius 1. Such circles are called unit circles. Note that a central angle of 1 radian intercepts an arc of length 1 on a unit circle.
Comparing Radians and Degrees
We know that the circumference of a unit circle is 
, so the central angle which spans the circle measures 
radians. Since the angle measure of the circle is 
, we know
radians, and thus 
radians and 
=1 radian.
Notation
We’ve written “radians” here to avoid confusion. In practice, we do not use a unit to indicate radian measure (because the definition of radian is linked to the unit used to measure the radius of the underlying circle). That is, we’ll say “the angle 
” in radians, but “the 
angle” when speaking in degrees.
Arclength
Radian measure yields a simple formula for arclength. If 
is a central angle of a circle of radius r, the length of the arc s intercepted by 
is 
.
