# 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. – MPath Mar 22 '18 at 11:09

## 2 Answers

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:

Apologies in advance for not using Mathjax and hope you understand my handwriting so here’s the algebraic proof:

In the first step the intent of the derivation is to transform into polar coordinates. So you want do substitute dx and dy. The polar coordinates are defined as written so you have to calculate the derivations of the coordinates. dx is then dependent on dr and dtheta as you have to make a total derivative. dx = dr cos theta - r sin theta dtheta and so on