The most area a fence can hold

A fixed length of fence, and a field to enclose with it. What shape holds the most?

The answer is a circle. It is one of the few results in mathematics that almost everybody would guess correctly, which makes it a poor advertisement for intuition and an excellent one for proof — because the guess is right, the reasons usually given for it are not quite proofs, and the gap between the two took two thousand years to close.

Every shape in that figure has the same perimeter and only the area changes. That is the only comparison the question allows: two shapes with different perimeters can be ranked by area for reasons that have nothing to do with their shape.

One number for the comparison #

Comparing at a fixed perimeter is awkward — the perimeter has to be chosen, and every answer is in units of it. The repair is a ratio that has no units at all: 4πA/L².

Both A and L² scale the same way when a shape is enlarged, so the ratio does not change with size. It is 1 for a circle, by the definitions of area and circumference, and the claim of the whole subject is that it is less than 1 for everything else.

Every figure on this page computes the ratio from coordinates — area by the shoelace formula on the drawn corners, perimeter by adding up the sides — and asserts that it never exceeds one. That assertion is the isoperimetric inequality, stated as a claim a figure would refuse to be drawn without.

More sides is always better #

The polygon sequence has a pattern with a short reason behind it. A regular n-gon of fixed perimeter has area growing with n, and the increase is checked at every step in the figure: 0.605, 0.785, 0.907, 0.977.

The reason is that a regular n-gon is n triangles meeting at the centre, each with base L/n and height the apothem. As n grows, the base shrinks in proportion while the height grows towards L/2π, so the total area climbs. The limit is the circle, and unrolling the circle into a triangle is exactly the n = ∞ case of the same decomposition.

That settles the competition among regular polygons and it settles nothing else. Irregular polygons, curved shapes and shapes with dents are all still in the running, and the question is which of them can be eliminated.

The other direction of the same fact #

An inequality relating two quantities can always be read twice, and the second reading is the one that gets used.

Fixing the perimeter and maximising the area gives the circle. Fixing the area and minimising the perimeter gives the circle as well, and it is the same statement: 4πA ≤ L² can be rearranged to say that a shape of area A needs a boundary at least √(4πA) long.

That second reading is why the inequality appears wherever a boundary is a cost and an interior is a benefit. A shape needs a certain amount of edge to hold a certain amount of middle, and the circle is the shape that gets away with the least. The soap film, the raindrop and the bubble are all sometimes offered as demonstrations — and this site does not use them, because they are physical systems with surface tension in them and the mathematical statement stands with every physical system deleted.

The purely mathematical reading is more useful anyway. It says that a region cannot have a small boundary and a large interior, which is a constraint on shapes rather than an observation about materials, and it is the form in which the inequality gets used to prove things about entirely different objects.

Every rectangle, swept #

Restricting to rectangles makes the problem one-dimensional and it can be settled by looking at all of them. The figure sweeps two hundred and forty aspect ratios and finds the maximum at the square — which it asserts, rather than deriving.

Two features of that curve are worth more than the answer.

It is flat at the top. A rectangle whose sides differ by ten per cent loses about a quarter of a per cent of its area. That is the general character of a maximum: near the top, the quantity is insensitive, which is why a roughly-square fence is nearly as good as a square one and why nobody notices the optimum in practice.

And it falls away steeply at the ends, to nothing at both. A rectangle twice as long as it is wide has lost an eighth. The penalty for being wrong is small until it is large.

Why symmetry is suggestive and not conclusive #

There is an argument for the circle that is even shorter than the dent argument and is worth examining because it is the one most people reach for.

The problem has no preferred direction: rotating a shape changes neither its area nor its perimeter. So the optimum, whatever it is, ought to be unchanged by rotation — and the only shape unchanged by every rotation is the circle.

The argument is appealing and it is not valid as stated. What the symmetry of a problem guarantees is that the set of optimal shapes is closed under rotation, not that any individual optimum is. If there were several optimal shapes, rotations could permute them and none need be round.

The gap is not hypothetical. Plenty of symmetric problems have unsymmetric answers — the way a symmetric arrangement of forces can buckle a column to one side, or the way a symmetric electorate can fail to have a symmetric winner. Symmetry arguments give the right answer here and they need the extra ingredient of uniqueness, which is exactly as hard to supply as existence.

Ruling out the dents #

Here is an argument that eliminates an entire class of shapes in one move.

Suppose a shape has a dent — a place where the boundary curves inward, so that a straight line between two boundary points passes outside. Reflect the dented piece of boundary across that line. The reflected piece has exactly the same length, since reflection is an isometry, so the perimeter is unchanged. And the shape now includes everything it did before plus the region between the line and the dent, so its area has strictly increased.

The figure does both computations. The perimeter is identical to within a billionth, which it must be, and the area rises by nearly forty per cent.

So no shape with a dent can be the best. Every non-convex shape is out, in one paragraph, without any calculation about which convex shape wins.

The move deserves a name because it recurs. Rather than searching for the best object, take an arbitrary candidate and show it can be improved — which eliminates it without identifying anything. A whole family of such improvements, each killing a class of candidates, narrows the field to whatever survives all of them. That is the same strategy as an infinite descent, where a supposed smallest counterexample is improved into a smaller one, except that descent ends in a contradiction and this ends in a survivor.

And the survivor is where the trouble starts, which is the next section.

The gap in the argument #

Steiner gave several arguments of that kind in the 1830s — the dent argument, an argument that the optimum must be symmetric about every direction, and others — and each eliminates a class of candidates. Together they seem to leave only the circle.

They do not prove the circle is the answer, and Weierstrass pointed out why. Every one of them has the form “if a best shape exists, it is not this one”. None of them establishes that a best shape exists at all.

That is not a pedantic objection, and the standard illustration is worth repeating: among positive numbers there is no smallest, and an argument of the form “if a smallest positive number exists, it is not 0.001, because 0.0005 is smaller” is entirely correct and proves nothing. Steiner’s arguments have that shape.

What makes the objection sharp rather than merely formal is that there are close relatives of this problem where no optimum exists. Ask for the shape of least area with a given perimeter and there is none: a shape can be made as thin as wanted, with area approaching nothing and never reaching it. Ask for the convex shape of given width with the least area and the answer is the Reuleaux triangle, which exists — but nothing in the shape of the question announced in advance which of the two behaviours to expect.

The gap is real and it was closed later, by arguments that establish existence first — usually by a compactness argument on the space of shapes, which is machinery that did not exist when Steiner was writing. It is a good example of a result being known, believed, defended by correct arguments, and unproved, all at once.

More sides, further out #

Pushing the polygon sequence out makes the convergence visible and puts a number on how quickly the circle is approached. A twenty-sided polygon of the same perimeter encloses 99.2% of the circle’s area; a hundred-sided one would reach 99.97%.

The shortfall shrinks like the square of the number of sides, which is fast, and it explains something about the practical version of this question. Anyone actually laying out a fence in straight sections gains almost nothing past a dozen of them — and the twelve-sided figure, at 0.977, is already within two and a half per cent of the theoretical best.

It also makes the circle’s status precise. It is not merely the best shape; it is the limit of a sequence of shapes each better than the last, none of which attains it. That is the same relationship the geometric series has with its own sum, and the same one Archimedes exploited to pin π between two polygons.

A round shape that is not the answer #

The circle is often described by the properties it has — perfectly round, the same width in every direction, rolls without wobbling — and a good test of whether those properties are what makes it optimal is to find a shape with some of them and check.

Curves of constant width are exactly that test. A Reuleaux triangle measures the same across in every direction and rolls smoothly under a plank, and it is not a circle. Its ratio is 0.897, which puts it between the hexagon and the pentagon — respectable, and nowhere near optimal.

So constant width is not what wins. Among shapes of constant width the circle is in fact the largest and the Reuleaux triangle the smallest, which is a separate theorem with its own history, and the fact that all of them have the same perimeter for a given width is Barbier’s theorem.

What the ratio does and does not measure #

The quotient 4πA/L² is often described as measuring how round a shape is, and it is worth being precise about what it actually detects, because the ranking has some surprises in it.

It is scale-free, so it cannot tell a shape from an enlarged copy. It is also unchanged by rotation and reflection. What it does see is how efficiently the boundary is spent, and that is not the same as roundness in any visual sense.

The L in the ranking scores 0.589, below the triangle’s 0.605, which is a fair verdict — the L has a great deal of boundary for its middle. But the ranking also puts the Reuleaux triangle at 0.897, above the pentagon, and a Reuleaux triangle has three visible corners while a pentagon has five. Corner count is not what the ratio measures.

A more instructive case: a very long thin rectangle and a very convoluted blob can have the same ratio while looking nothing alike. The number is a single summary and it compresses a shape to one figure, which is exactly its usefulness and exactly its limit — the same trade the determinant makes when it compresses a map into one number.

What the picture cannot show #

Every shape drawn is a polygon or a circle. The competition is over all closed curves, including curves with no corners anywhere, curves that wiggle at every scale, and curves of infinite length. Nothing here draws one, and the inequality’s proof has to cover them.

The sweep of rectangles is a sweep of rectangles. Two hundred and forty of them is exhaustive within its class and says nothing about any shape outside it. The figure is honest about that in the only way available — by naming its class in its own title.

And the existence of an optimum is nowhere visible. Every figure compares finitely many shapes and reports which of them did best. That there is a best shape at all, among infinitely many candidates, is exactly the thing the pictures cannot supply and exactly the thing Steiner’s arguments assumed.

Where the ladder goes next #

Upwards, the same question in three dimensions gives the sphere, and the proof is harder in the way such things usually are. Downwards in generality, the same question with a constraint — the largest area enclosed by a fence of given length with one side along a straight wall — has a different answer, a half-circle, and the reflection trick that produces it is a small classic.

Sideways, the inequality has a discrete cousin worth knowing: among lattice polygons with a given number of boundary points, which encloses the most? That is a question about counting dots rather than about measuring, and its answer is not a circle, because the lattice has no circles in it.

And there is the reverse question, which is where the subject gets its modern shape. Given that a shape’s ratio is close to 1, must the shape be close to a circle? The answer is yes, with a quantitative bound, and results of that form — nearly optimal implies nearly the optimum — are the version of the isoperimetric inequality that turns out to be useful, because in practice nothing is ever exactly optimal.