How to find the arc length between any two points (real numbers) on the circumference of a circle with center at the origin?

Question

Suppose I'm given two points: (x1, y1) and (x2, y2) (which are real numbers) lying on the circumference of a circle with radius r and centred at the origin, how do I find the arc length between those two points (the arc with shorter length)?

Answer 1

We have the following situation:

Points $A$ and $B$ are given by their coordinates. So you can calculate their distance. Now, we have an isosceles triangle with given side lengths. We can calculate $\alpha$.

Given $\alpha$ and $r$, the arclength can be calculated.

Answer 2

Find the distance $d$ between these two points. Your desired arc $\alpha$ is obtained from solving $\frac{\pi}{\alpha} = \frac{D}{d}$, where $D = 2r$ is the diameter of the circle