Arc Length and Surface Area with Polar Coordinates

The arc length of a polar curve is simply how long the curve is, measured by adding up lots of tiny steps along it as the angle turns. As before, "arc" just means a piece of curve, not only a piece of a circle. It's good for measuring spirals and loops, like the track on a CD or the groove on a record, which are kilometers long. The formula $\sqrt{r^2 + (dr/d\theta)^2}$ looks heavy, but it's just two moves combined: $r$ (how far you slide around the origin per bit of turning, since a bigger circle means more distance) and $dr/d\theta$ (how fast you're drifting outward). Those two moves are at right angles, so Pythagoras glues them into one step, and the integral adds up the steps. It's the same arc length idea you already know, just measured in turns instead of $x$.

Where it comes from. You don't need a new idea here. In the polar tangents lesson we saw that a polar curve is a parametric curve with $x = r\cos\theta$ and $y = r\sin\theta$. So use the parametric arc length formula. Differentiating with the product rule, squaring and adding,

after the cross terms cancel and $\cos^2\theta + \sin^2\theta = 1$ cleans up the rest.

That tidy answer isn't an accident, and it's worth seeing why. When $\theta$ turns a little, the point moves around the origin by an arc of length $r\,d\theta$ (that's the arc-length-of-a-circle formula, angle times radius). At the same time it moves outward by $dr$. Moving around and moving outward are at right angles to each other, so the tiny step is the hypotenuse $\sqrt{(r\,d\theta)^2 + (dr)^2}$. Pull out $d\theta$ and you have the formula.

Surface area. Same recipe as always: $S = \int 2\pi y\,ds$ about the x-axis, with $y = r\sin\theta$ and $ds = \sqrt{r^2 + (dr/d\theta)^2}\,d\theta$.

The groove on a vinyl record and the track of a CD are spirals, and their total length (several kilometers on a CD) is a polar arc length.