What is a Radian?
When an object moves in a circular path about some axis we can see that the arc which it travels is proportional to the angle it subtends. If the object completes one whole path of the circumference of the circle its path length is 2 π r.
In the diagram the arc length s can be calculated by $s=\frac{2 \pi}{360^{^{\circ}}}r\theta$ where θ is in degrees. This is a fairly cumbersome equation, and can be simplified if we use radians instead of degrees. A radian is equal to the angle subtended by an arc of length equal to one radius and approximates to 57.3° (3 s.f.) and there are 2π radians in one complete circle so:
$$\large 2\pi (\mathrm{radians}) = 360^{\circ}$$and an angle measured in radians can be related to one measured in degrees by the equation:
Therefore measuring our angles in radians results in the much simpler equation:
$$\large s=r\theta$$Where θ is in radians.
Therefore measuring our angles in radians results in the much simpler equation $s=r\theta$ where θ is in radians.