A circle unrolled into a triangle

Everyone is told that the area of a circle is $\pi r^2$. Almost nobody is told why, and the formula does nothing to suggest it. A radius is a length; squaring it gives an area; the factor $\pi$ appears from somewhere and is left there.

The picture is a dissection, in the same spirit as two squares and four triangles, and it takes about one sentence to describe. Cut the disc into concentric rings. Straighten each ring out. Stack them.

Why the stack is a triangle #

Take the ring at radius $s$, a hair thick. Cut it once and pull it straight and it is a strip of length $2\pi s$ — its own circumference — and the same hair’s thickness.

That length is proportional to $s$. The ring at the very centre has length nothing; the ring at the outer edge has length $2\pi r$; and every ring in between has a length that grows in exact proportion to how far out it was.

So stack them with the shortest at the top and the longest at the bottom, each directly above the next. The left ends line up, the right ends line up, and the profile between them is a straight line, because the lengths are proportional to the height in the stack. A shape with a flat base of $2\pi r$, an apex, and straight sides is a triangle.

Its area follows immediately. A triangle is half its base times its height, and here the base is the circumference $2\pi r$ and the height is the radius $r$:

$$\text{area} = \tfrac{1}{2} \cdot 2\pi r \cdot r = \pi r^2.$$

The $\pi$ was never mysterious. It walked in with the circumference and stayed, and the $r^2$ is one $r$ from the circumference and one from the radius. Two lengths multiplied make an area, which is the only thing an area was ever going to be.

The staircase, and the limit that removes it #

Now the honest part, because the paragraph above skipped a step that the picture makes very easy to skip.

A ring is not a hair thick. It has an inner radius and an outer one, so straightened it is not a rectangle but a very slightly tapered strip — longer along its outer edge than its inner. Stacked, they do not make a triangle. They make a staircase.

This is exactly the situation of rectangles under a curve, and it deserves the same treatment. The staircase is wrong by an amount that shrinks as the rings get thinner, and the triangle is defined as what the staircases converge to rather than as something any of them equal. No number of rings makes the picture correct. The limit makes the claim correct, and the pictures are stages on the way to it.

Exactly how wrong #

The error is not merely small. It has a formula, and the formula is unusually clean, so there is no excuse for leaving the reader with an impression where a number will do.

Each strip in these figures is drawn at its ring’s outer circumference, which is the generous choice — a ring is being replaced by a rectangle as long as its longest edge. The strip at position $k$ out of $n$ therefore has length $2\pi r \cdot k/n$ and thickness $r/n$, and the whole stack has area

$$\sum_{k=1}^{n} 2\pi r \frac{k}{n} \cdot \frac{r}{n} = \frac{2\pi r^2}{n^2} \cdot \frac{n(n+1)}{2} = \pi r^2\left(1 + \frac{1}{n}\right).$$

So the stack of strips overshoots the disc by exactly $1/n$ of it. At the twelve rings of the opening figure that is $8.3\%$; at twenty-four it is $4.2\%$; at forty, $2.5\%$. The triangle outlined over them has area exactly $\pi r^2$, and the strips inside it stick out past its sloping side by precisely that fraction.

Two things follow. The first is that the error falls like $1/n$, which is slow — the same rate as left-endpoint rectangles under a curve, and for the same reason, since this is that construction with the strips lying down. Ten times the rings buys one decimal place.

The second is that the sum used to get there — $1 + 2 + \cdots + n = n(n+1)/2$ — is the two interlocking staircases, which is a figurate-number fact doing load-bearing work inside a statement about circles. The $+1$ in $n(n+1)$ is the entire error.

That last figure is the one to be careful about. It looks like a triangle, and looking like a triangle is not the argument — it is the thing the argument has to establish. A reader who accepts the picture at twenty-four rings has accepted it for the wrong reason, and will accept the next plausible-looking staircase too.

What the argument is quietly assuming #

There is a second gap, and it is more interesting than the first because no amount of refinement closes it.

The proof uses the circumference. It says the outer ring has length $2\pi r$, and everything else follows. So it has already assumed that the ratio of circumference to diameter is $\pi$ — that is where the constant enters, and the derivation is not a computation of $\pi$ but a conversion of one fact about $\pi$ into another.

Which is fine, and worth being explicit about, because the relationship it establishes is genuinely substantial. Two quantities are attached to a circle: the distance round it, and the space inside it. There is no obvious reason the same constant should govern both. That $C = 2\pi r$ and $A = \pi r^2$ share a $\pi$ is a theorem, and this dissection is its proof.

Stated that way the picture says something sharper than the area is $\pi r^2$. It says: the area is half the circumference times the radius, whatever the circumference happens to be. That form makes no reference to $\pi$ at all, needs no constant, and is where the content actually lives. The $\pi$ version is what is left after substituting a definition.

The move has a name worth knowing, because it recurs: the constant is being defined at one place in the chain and derived everywhere after it. $\pi$ is defined by the circumference here and shows up in the area as a consequence. Define it by the area instead and the circumference becomes the consequence. Neither is more fundamental; what is not available is defining it twice and calling their agreement a discovery — which is the same circularity that makes a trigonometric proof of the Pythagorean theorem difficult to arrange honestly.

It also explains why the same relation reappears in odd places. The surface area of a sphere is four times the area of its great circle, and its volume is a third of the surface times the radius — the same half-base-times-height move one dimension up, with a third in place of a half because the cone of thin pyramids has three dimensions to divide by rather than two.

Archimedes did it differently, and had to #

The dissection is old in spirit and modern in its casualness about limits. Archimedes proved the same result around 250 BC, and his version is longer for a reason worth understanding.

He had no way to say “the staircases converge”. What he had was the method of exhaustion, which proves an equality by ruling out both inequalities. To show that the disc’s area equals the triangle’s, he assumed it was greater, inscribed enough polygons to force a contradiction, then assumed it was less, circumscribed enough to force another, and concluded that it was neither.

That is a proof by double contradiction and it is entirely rigorous. It is also completely unilluminating about why — it establishes that no other answer survives, without ever showing where this answer comes from. The dissection is the opposite: it makes the answer obvious and leaves the rigour to be supplied later.

It is worth adding that the same man, with the same method, produced the first good numerical bound on $\pi$ — inscribing and circumscribing regular polygons of ninety-six sides to get $3\tfrac{10}{71} < \pi < 3\tfrac{1}{7}$, which pins it between $3.1408$ and $3.1429$. That is two correct decimal places, obtained by hand, from a polygon nobody could draw legibly, and it stood as the best available value in Europe for something like eighteen centuries. The exhaustion method is not merely a rigorous substitute for the limit; it is a computational technique, and it was the one that worked.

Which is the recurring trade in this collection. The picture explains and does not prove; exhaustion proves and does not explain; and the modern limit, which arrived two thousand years after Archimedes, is what lets one argument do both. It is worth remembering that the interval was two thousand years, and that the missing piece was not cleverness but a definition.

The other dissection #

There is a second, better-known cutting of the same disc, and comparing the two is instructive.

Instead of rings, cut the disc into thin wedges like a pie, then lay them out alternately point-up and point-down. They interlock into something very close to a rectangle, of height $r$ and width $\pi r$ — half the circumference, since half the wedges point each way. Its area is $\pi r \cdot r = \pi r^2$.

The wedge version has an advantage and a disadvantage against the rings. The advantage: its error is easier to see, since the top and bottom edges are visibly scalloped rather than straight and nobody is tempted to think the shape is already a rectangle. The disadvantage: the wedges are not straightened, so the argument needs the reader to accept that a thin wedge is nearly a triangle, which is the same limit in a less obvious place.

Neither is more rigorous than the other. Both are the same theorem with the same gap, plugged by the same limit, and the choice between them is about which approximation a reader is more likely to notice — which is a real consideration and not a mathematical one.

What the picture cannot show #

Every figure here has finitely many rings and the claim is about infinitely many. The gap is not decoration: it is where the entire content of the word area is hiding, and no drawing closes it.

The particular deception is that the pictures get better looking as they get closer to the claim, so a reader’s confidence rises for a reason unconnected to the argument. At forty rings the staircase is invisible and the shape is still not a triangle. The moment the eye stops objecting is not the moment the mathematics stops needing an argument, and those two moments are far apart here — which is precisely what makes this a good place to notice the difference.

The picture also cannot show what happens if the ambient geometry is not flat. On a sphere, a circle of radius $r$ — measured along the surface — has an area strictly less than $\pi r^2$, and the shortfall is a measure of the curvature. The rings still exist and are still concentric; what fails is that a ring at distance $s$ has circumference $2\pi s$, because on a curved surface it does not. The flatness of the paper is an assumption of the proof, and the paper cannot report it, in exactly the way a dissection cannot report that it is Euclidean.

The ladder from here #

Rungs above: the wedge dissection given its own figure and its error bounded. Archimedes’ exhaustion argument, drawn as inscribed and circumscribed polygons closing on the circle from both sides — the first place where a quantity was pinned between two sequences. The circumference itself, and the fact that $\pi$ has to be defined somewhere in the chain rather than derived everywhere. The sphere’s surface and volume by the same slicing. Cavalieri’s principle, and the paradox that makes it dangerous. The isoperimetric theorem, which says the circle is the shape enclosing the most area for its perimeter — the deep version of what this essay computes. Polar coordinates and the integral $\int_0^r 2\pi s \, ds$, which is this dissection with the limit written down. And $\pi$ computed rather than assumed, where needles on a floor turn out to be one of the worst available methods.

What the ring is really for #

The move that makes this work is worth naming because it is the whole of integration in one gesture: slice the region so that each slice is a shape whose measure is already known, and along a direction in which that measure varies simply.

Rings are the right slices for a disc because a ring’s length depends only on how far out it is, and depends on it linearly. Rectangles are the right slices under a curve because a rectangle’s area depends only on the height there. The wedges of a circle seen as a turning radius are the right slices for a sector, because a sector’s area depends only on its angle. Choose the slices badly and the same region becomes intractable — cut the disc into vertical strips instead and each strip’s height is $2\sqrt{r^2 - x^2}$, which is correct, is not linear in anything, and needs a trigonometric substitution to sum.

The picture is a good picture because the slicing was well chosen. That choice is the mathematical content, and it is the part a finished diagram makes look inevitable.