What is Arc length?

Example 18.8 (Archimedes and the inscribed polygons)

For the unit circle, the inscribed regular $n$-gon has length $L_n = 2n\sin\frac\pi n$, and the expansion $\sin x = x - \frac{x^3}6 + O(x^5)$ gives

the polygonal lengths of Theorem 18.7 converge quadratically. Numerically: $L_6 = 6$ (the hexagon, giving the crude $\pi > 3$), while $L_{96} = 192\sin\frac{\pi}{96} \approx 6.28206$ against $2\pi \approx 6.28319$ — the error $0.00113$ agrees with the predicted $\pi^3/(3\cdot96^2) \approx 0.00112$. This is why Archimedes, doubling the hexagon five times to $96$ sides, could bracket $\pi$ to three digits by hand: each doubling divides the error by four. The supremum in the polygonal characterization is not just attained in the limit; it is attained fast, because a smooth curve separates from its chords only at second order.

Example 18.11 (Circle and helix)

For the circle $\gamma(t) = (R\cos t, R\sin t)$, $\norm{\gamma'} = R$, so $s = Rt$ and the length of a full turn is $2\pi R$. For the helix $\gamma(t) = (a\cos t,\ a\sin t,\ bt)$ with $a > 0$, $\norm{\gamma'(t)} = \sqrt{a^2 + b^2}$ is constant: the helix is traversed at constant speed, and $s = t\sqrt{a^2 + b^2}$.

Example 18.12 (Arc length in polar coordinates)

A polar curve $r = r(\theta)$ is the arc $\gamma(\theta) = (r\cos\theta,\ r\sin\theta)$, with

(the cross terms cancel): the polar length element is $\sqrt{r^2 + r'^2}\,\dd\theta$. For the cardioid $r = 1 + \cos\theta$:

and on $\intcc{-\pi}{\pi}$ the half-angle cosine is nonnegative, so

like the cycloid arch of Exercise 18.1, a curve built from circles has a rational length, with no $\pi$ anywhere. The half-angle factorization is the standard trick for lengths of circle-generated curves; when it fails (the ellipse), the length is a genuinely new function — an elliptic integral, beyond elementary closed forms.