Consider the shape r=θ
At a certain angle θ from the origin, the length of the radius is θ and the angle is incremented by dθ, resulting in the second side being θ+dθ in length.
What is the correct approximation of the third side? Also what is the precise value of the alpha angle? Or is there a function to predict its value at θ?
For instance, the differential element of a circle is:
The third side is rdθ in length
You have the polar function, $f(\theta) = \theta $
The differential length of the graph of this function is given by
$ (ds)^2 = f^2 d\theta^2 + ( d f) ^2 $
So that the length of the curve between $\theta_1 $ and $\theta_2 $ is
$ L = \displaystyle \int_{\theta_1}^{\theta_2} \sqrt{ f^2 + \left(\dfrac{df}{d\theta} \right) ^2 } d \theta $
But in differential form (which is your question), the differential length is
$ d s = \sqrt{ f^2 + \left( \dfrac{df}{d\theta} \right)^2 } d \theta = \sqrt{ \theta^2 + 1 } \ d\theta $