The 5-Kilometer Track on a CD

A CD stores data along one spiral track that winds outward about 20,000 times. Its length is a polar arc length integral, and with a track pitch of 1.6 micrometers it comes to over five kilometers on a disc you can hold in your hand.

How it works

Arc length with polar coordinates

In one sentence

The length of the polar curve $r = f(\theta)$ for $\alpha \le \theta \le \beta$ is $L = \int_\alpha^\beta\sqrt{f(\theta)^2 + f'(\theta)^2}\,d\theta$ .

General formula

{\htmlData{sym=0}{r(\theta)}} = {\htmlData{sym=1}{r_0}} + \frac{\htmlData{sym=3}{{p}}}{2\pi}{\htmlData{sym=4}{\theta}}, \qquad {\htmlData{sym=5}{L}} = \int_0^{\Theta}\sqrt{{\htmlData{sym=0}{r(\theta)}}^2 + \left(\frac{dr}{d{\htmlData{sym=4}{\theta}}}\right)^2}\,d{\htmlData{sym=4}{\theta}} \approx \frac{\pi\left({\htmlData{sym=2}{R}}^2 - {\htmlData{sym=1}{r_0}}^2\right)}{\htmlData{sym=3}{{p}}}

Each small step along the curve moves around the circle ( $r\,d\theta$ ) and outward ( $dr$ ); the Pythagorean theorem combines them.

What the symbols mean

$r(\theta)$
Track radius
Distance of the track from the center at angle θ.
$r_0$
Inner radius
Where the track starts.
$R$
Outer radius
Where the track ends.
$p$
Pitch
Gap between neighboring turns.
$\theta$
Angle
Total angle wound so far.
$L$
Track length
Length of the whole spiral.

The track is an Archimedean spiral: each turn moves the radius out by the pitch $p$ , so $r = r_0 + \frac{p}{2\pi}\theta$ .
Polar arc length adds $ds = \sqrt{r^2 + (dr/d\theta)^2}\,d\theta$ along the curve.
Here $dr/d\theta = p/2\pi$ is tiny next to $r$ , so $ds \approx r\,d\theta$ and the integral becomes the area of the ring divided by the pitch.
For a real CD ( $r_0 = 25$ mm, $R = 58$ mm, $p = 1.6\ \mu$ m) that's about 5.4 km.

Learn the math

Arc Length and Surface Area with Polar CoordinatesCalc II · Lesson 23, Parametric Equations and Polar Coordinates