Consider an arc of length l in a circle of radius r (help yourself with a simple diagram of a circle and its arc).
The arc produces an angle theta at the centre.
Radian measure of the angle theta is expressed as
Angle in radians = length of the arc, l / (radius, r)
Measuring arc length and radius due to a every angle is not so trivial. Thus expression of angle in radians would have been too cumbersome.
However, there is a way out.
If we make the arc a semicircle, the length of the arc is equal to semiperimeter of the circle, i.e. 2*pi*r/2 = pi*r.
Therefore, angle in radians = pi*r / r = pi.
But the angle subtended at the centre by a semicircle is 180°.
So, pi radians = 180°
1 radian = (180/pi) °
This can be approximated to 57.3 degrees, but there remains an element of approximation still, as the value of pi itself is approximated, no matter how many digits after the decimel is taken.
So, we found that without having to measure the arc length or radius of the circle, any angle in degrees can be expressed in radians easily.
Furthermore, it is clear that expression of radians in terms of pi is more authentic than in terms of approximated digits.