Reuleaux Polygons: Shapes as Round as a Circle, Without Being One

Constant width without being a circle

A shape has constant width if the distance between two parallel supporting lines touching it stays exactly the same no matter which direction you measure — the way a circle's diameter is the same in every direction. The surprise is that the circle is far from the only such shape. A Reuleaux polygon is built from a regular polygon with an odd number of sides (3, 5, 7, 9, 11…): replace each side with a circular arc, centred at the opposite vertex, with radius equal to the polygon's width. The simplest and most famous case, built on a triangle, is the Reuleaux triangle, but the same construction works for any odd-sided regular polygon — it fails for even-sided ones because they have no single vertex directly opposite each edge.

Why odd sides are non-negotiable

The construction relies on every edge having exactly one vertex diametrically opposite it, so that a single arc, centred there, can bulge outward and keep the width constant across that edge. A regular polygon with an even number of sides has each vertex opposite another vertex, not the midpoint of an edge, so the opposite-vertex arc trick simply has no valid centre to use — which is exactly why every curve of constant width built this way comes from an odd-sided base polygon, and why a 'Reuleaux square' or 'Reuleaux hexagon' in this specific sense does not exist (though other, differently-constructed constant-width shapes built from even-sided bases do exist, using a more general arc arrangement).

Barbier's theorem: the perimeter doesn't care about the shape

Every curve of constant width w — the circle, the Reuleaux triangle, the Reuleaux pentagon, and infinitely many other, less symmetric constant-width shapes — has exactly the same perimeter, πw, regardless of its specific shape. This is Barbier's theorem (1860), and it is genuinely surprising: a Reuleaux triangle of width w looks nothing like a circle of diameter w, yet a piece of string exactly long enough to wrap once around either shape is exactly the same length, πw, every time.

perimeter of ANY curve of constant width w  =  pi * w   (Barbier's theorem)

circle, diameter w:            perimeter = pi*w         (obviously)
Reuleaux triangle, width w:    perimeter = 3 * (w * pi/3) = pi*w   (three arcs, each 1/6 of a full circle,
                                each arc spans 60 deg = pi/3 rad of a circle radius w, arc length = w*pi/3)

Rolling under a flat plate: a shape, not a wheel

Constant width does not mean constant radius — the Reuleaux triangle's corners are noticeably farther from its centroid than its arc midpoints are, so unlike a circle it wobbles as it rolls, and its centre traces a small closed curve rather than a straight line. What constant width actually guarantees is that a flat plate resting on top of the rotating shape stays at a fixed height throughout the roll, always touching at exactly one point with the same overall clearance. This is precisely the mechanical property exploited by Harry Watts' 1914 square-hole drill bit — a Reuleaux triangle rotated inside a square guide traces out a shape that is almost, but not quite, a perfect square, with slightly rounded corners, because the triangle's own centre must wander during the rotation to keep all four sides of the guide simultaneously in contact.

Where Reuleaux shapes actually get used

Beyond the square-hole drill, constant-width curves solve a specific manufacturing and design problem: a mechanism needs to rotate freely inside a tightly fitting guide without ever jamming or leaving a gap. British 20-, 50- pence and Canadian dollar coins use curves of constant width (technically Reuleaux-heptagon-derived shapes) rather than circles specifically so vending machines can validate their diameter from any orientation while still letting the coin look visibly non-circular to the eye — a circle-only vending mechanism cannot tell a genuine coin from a slightly-wrong-sized disc as reliably, since diameter alone is ambiguous, but distinct multi-sided constant-width shapes keep a consistent measured width while remaining visually and tactilely distinguishable from each other.

Frequently asked questions

Try it live

Everything above runs in your browser — open Reuleaux Polygons and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.