How long is a curve?

A ruler measures straight segments, and a circle contains none: no piece of it, however short, is straight. Part 2 leaned on "circumference" anyway. Before that debt is settled, length itself needs a definition that works on curved lines.

Measure it with straight pieces

Mark points along the curve in order, join consecutive marks by straight segments, and total their lengths. Polygonal length and rectifiability makes this precise: a path is a continuous map $\gamma :[a,b]\to {\mathbb{R}}^n$, every choice of marks gives an inscribed polygonal length, and the arc length is the supremum over all choices. A chord is never longer than the arc it cuts (Every endpoint chord is no longer than the arc: $\parallel \gamma (b)-\gamma (a){\parallel }_2\le L(\gamma )$), and adding a mark never shortens a total, so refinement only pushes the totals up. When they stay bounded, the path is called rectifiable, and its length is a real number.

The definition can refuse

The length is a supremum, and a supremum can be infinite. The Koch curve is the classic refusal: polygonal paths of length $(4\mathrm{/}3{)}^n$ converge uniformly to a path whose inscribed totals grow without bound, so it has no length. It is built to fail, but ordinary-looking curves fail too: the graph of $x\sin (1\mathrm{/}x)$ is the graph of a continuous function and is not rectifiable either.

So "the circumference of a circle" is a claim. Someone has to prove that the circle's inscribed totals stay bounded, or the phrase names nothing.

The circle pays up

It does stay bounded, and the supremum is an old friend. The upper unit semicircle $\gamma (t)=(\cos t,\sin t)$, $0\le t\le \pi$, is rectifiable with arc length exactly $\pi$ (The arc length of a unit semicircle is pi): the same number Part 2 defined as twice the first positive zero of cosine, recovered here as a length. In general the circumference of a circle of radius $r$, defined as the arc length of the path tracing it (Circular arcs, circumference as arc length, and diameter), is $2\pi r$ (Every circle has circumference 2 pi r and circumference-to-diameter ratio pi).

The question is finally well posed

Part 1 defined area. Part 2 defined π. This part defined length and paid circumference's debts. Every word in "the area of a circle of radius $r$ is $\pi r^2$" now has a meaning, and every meaning is proved. What remains is the theorem itself. Next: the proof.