This is a very nice result: the length of an arc of a circle is equal to the radius times the angle of that arc.
This is, actually, how angle is usually defined, but here we derived it from the starting point that a circle is defined by $x = R \cos(\theta)$ and $y = R \sin(\theta)$.
Now @heather, how much arc length do you get if the angle goes from 0 to $2\pi$?
You actually just did some multivariable calculus.
Remember we started with the Pythagorean thingy $ds = \sqrt{dx^2 + dy^2}$.
That tells you the length of a bit of curve in the 2D plane.
Then you used a parametrization of the circle, i.e. $x = R \cos(\theta)$ and $y = R \sin(\theta)$ to figure out $dx$ and $dy$ in terms of a single variable $\theta$, and then you did the integral to get the total path length.
