# Derivation of $d\theta = ds/r$

I was reading about Uniform Circular motion and I came across this formula: $d\theta = ds/r$. ($r$ being the radius, $d\theta$ being the angle swept by the radius vector and $ds$ being the arc length)

I thought that the formula is basically the definition of radian measure. But deeper research led me to the following derivation:

What is $dr$ and how did we get $ds^2 = dr^2 + r^2 d\theta^2$?

I know this is very basic but there was no image to represent this pictorically and I couldn't get much results googling these formulae.

• I prefer to rearrange the formula as radius * theta = arc_length (typing on mobile...). If theta is 2 * pi, then the arc_length is the circumference. If theta is any other number, you can consider the total arc_length to be some constant times the circumference.
– user140374
Commented Mar 22, 2018 at 11:09

This is a formula used to find the arc lengths swept in polar-coordinates. A geometrical proof is as follows: Taking a very small section of a curve we get the approximation of one side being $rd\theta$ and the other side being $dr$ so the arc length is approximately equal to the hypotenuse and by Pythagoras Theorem we get the expression: