Sixty Metres of Fence Encloses More Ground the Rounder You Make It

Why the hexagon wins in practice

The circle is the best shape for one enclosure and a poor one for many, because circles cannot be packed without leaving gaps. Among the shapes that tile a plane with no waste — triangle, square, hexagon — the hexagon is the roundest, and it therefore encloses the most area per unit of shared wall. That is the argument behind honeycomb, behind hexagonal paving, behind the shape of mesh in a garden screen, and behind hex bolts, whose six flats give a spanner more purchase than four would while keeping the head compact. When a project involves one boundary, aim round; when it involves repeated cells, aim hexagonal.

Diminishing returns after about eight sides

The first few sides do almost all the work. Going from three to four buys eighteen percentage points of the circle's area, four to six buys twelve, six to twelve buys seven, and everything beyond twelve buys the last two and a bit. Since each extra side costs a corner post, a cut and a joint, the return on complexity collapses long before the geometry does — which is why real enclosures are square when the material is cheap and hexagonal or octagonal when it is not, and almost never twenty-sided. The same reasoning appears wherever a curve is approximated by straight segments, from a garden bed to a rendered arc on a screen.

Frequently asked questions

Is the circle really the best possible shape?

Yes, and it is a theorem rather than an observation. Among all closed curves of a given length, the circle encloses the greatest area — the isoperimetric inequality, proved in the nineteenth century after being suspected since antiquity. No irregular shape, however clever, beats it, and no polygon reaches it. The result also runs the other way: for a given area, the circle has the shortest boundary, which is why a soap bubble is round and why anything minimising surface for a fixed content ends up spherical.

What if the plot is not free to be any shape?

Then the constraint usually beats the geometry, and it is worth saying so. A boundary that must follow an existing wall, a right of way or a straight property line takes the choice away, and a shape that wastes area on paper may be the only one that fits. The calculation stays useful as a ceiling: it tells you what the same length of fence could have enclosed, which is the number to weigh against the cost of the alternative. Where one side is an existing wall the problem changes shape entirely — with three sides of fence against a straight wall, the best rectangle is twice as wide as it is deep, and that is worth checking before cutting anything.