Reuleaux Triangle

What a Reuleaux triangle is

Take an equilateral triangle. Put a compass point on one corner, open it to the length of a side, and sweep an arc connecting the other two corners. Do the same from each of the three corners. The three bulging arcs join into a curved triangle — and that shape, the Reuleaux triangle, has a property that feels impossible the first time you see it: it is exactly the same width in every direction, just like a circle.

"Width" here has a precise meaning. Trap the shape between two parallel lines (think of the jaws of a calliper) and squeeze until both lines touch. The gap between them is the width in that orientation. For a circle the gap is always the diameter, no matter how you turn it. For most shapes — a square, an ordinary triangle, an egg — the gap changes as you rotate. For a Reuleaux triangle it does not. Rotate it through any angle and the calliper still reads the same number. Shapes with this property are called curves of constant width, and the Reuleaux triangle is the simplest one that is not a circle.

It is named for Franz Reuleaux, the 19th-century German engineer often called the father of kinematics, who used it as a teaching example in machine design. The shape itself is far older — Leonhard Euler studied constant-width curves in the 1700s — but Reuleaux's name stuck to the three-arc triangle.

The geometry: arcs, width, area, perimeter

Everything about the shape follows from one rule: each arc is a circular arc of radius equal to the side length, centred on the opposite vertex. Because the radius equals the side, the point on any arc directly opposite a vertex is exactly one side-length away from that vertex — and that is what makes the width constant. Whatever direction you measure, you are always measuring from a vertex (a sharp corner) straight across to a point on the opposite arc, a distance fixed at the radius s.

Let s = side length of the underlying equilateral triangle = arc radius.

  Width            w   = s          (constant in every direction)
  Each arc spans   60° = π/3 rad
  Perimeter        P   = 3 · (s · π/3) = π·s
  Area             A   = ½(π − √3)·s²  ≈ 0.70477·s²
  Corner angle     each vertex is an exterior corner of 120° interior

Two of those results are genuinely remarkable. The perimeter is exactly π·s — the same as the circumference of a circle whose diameter is s. That is not a coincidence of this shape; it is Barbier's theorem, which says every curve of constant width w has perimeter π·w. A Reuleaux triangle, a circle, and a wobbly constant-width blob of width 1 metre all have the same 3.1416-metre perimeter.

The area, by contrast, is the smallest possible. Among all curves of constant width w, the Reuleaux triangle encloses the least area — that is the Blaschke–Lebesgue theorem — while the circle encloses the most. So a Reuleaux triangle is the leanest, "pointiest" constant-width shape you can draw:

For the same width w:

  Reuleaux triangle area  = ½(π − √3)·w²  ≈ 0.7048·w²   (minimum)
  Circle area             = (π/4)·w²      ≈ 0.7854·w²   (maximum)

  Reuleaux is ~10.3% smaller in area than the circle of equal width.

Why it rolls smoothly — and why it is not a wheel

Constant width is exactly the property you need to roll a load smoothly. Lay several Reuleaux-triangle rollers under a flat board and push: because the distance from the ground to the top tangent never changes, the board stays at a constant height and glides forward without bobbing — precisely as it would on round rollers. Ancient movers used round logs for the same reason; Reuleaux rollers would work just as well.

So why are wheels round and not Reuleaux? The catch is the centre. As a Reuleaux triangle rolls, its centroid rises and falls. The shape's top stays level, but its middle does not. A wheel is bolted to an axle through its centre, and if that centre bobs up and down, the whole vehicle bounces. A circle is the only constant-width curve whose centre also stays at a fixed height while rolling, which is why axles are round. Constant width buys you a smooth top; only the circle gives you a smooth centre as well.

This distinction is the single most common misconception about the shape. "A Reuleaux triangle can be a wheel" is half true: it can carry a load on free rollers, but it cannot be mounted on a hub.

Drilling a square hole: the Watts drill

Spin a Reuleaux triangle inside a square frame whose side equals the triangle's width, and something strange happens: the three corners stay pressed against all four walls and the shape sweeps out nearly the whole square. The corners trace almost-straight lines; only four tiny patches near the square's corners are never reached.

In 1914 the Watts Brothers Tool Works of Wilmerding, Pennsylvania turned this into a real drill bit. A Reuleaux-shaped cutter is held in a floating chuck — a guide template plus an Oldham-style coupling that lets the bit's centre wander along the path the geometry demands rather than spinning about a fixed axis. As the bit turns, it bores a hole that is about 98.8% of a true square. What is left out is four small rounded corners (each a short elliptical arc, not a circular fillet) that together amount to only about 1.2% of the square's area. For mortises, square sockets, and broaching-style work where perfectly sharp internal corners are not needed, that is close enough — and far faster than chiselling corners by hand.

Quantity	Value (width w = 1)	Note
True square area	1.000 w²	The target hole
Area actually cut	≈ 0.9877 w²	≈ 98.8% coverage
Uncut corner regions	≈ 0.0123 w² total	Four rounded slivers, ~1.2% of square area
Centre-of-bit motion	Small near-elliptical orbit	Why a floating/Oldham chuck is required

The same idea scales to other polygons. A Reuleaux pentagon spun in the right frame approximates a regular pentagon's hole; in general, an odd-sided Reuleaux polygon of width w can rough out the corresponding regular polygon. The square is the famous case because square mortises and sockets are so common.

Where Reuleaux shapes show up

Application	Why a constant-width / Reuleaux shape
Watts square-hole drill	Reuleaux rotor sweeps ~98.8% of a square; floating chuck guides the centre path
Wankel rotary engine rotor	Curved-triangle rotor (Reuleaux-inspired) sweeps three chambers in one epitrochoidal housing
UK 50p and 20p coins	Reuleaux heptagons — constant width gauges the same in any orientation, distinct by touch
Guitar picks, some pencils	Triangular feel with no sharp jam point; rolls without snagging
Film-advance / Geneva-style mechanisms	Reuleaux-rotor "movie-camera" intermittent drives used the shape's geometry
Architectural windows & vaults	Gothic and modern facades use the three-arc curve for its even, "fat-triangle" look
Roller conveyors (constant-width rollers)	Loads ride level on non-circular rollers; demonstration and teaching rigs

The Wankel deserves a caveat. Its rotor is descended from the Reuleaux triangle, but it is not a pure three-arc curve: the flanks are bulged a little more (or relieved with combustion pockets) so the apex seals can follow the engine's epitrochoidal chamber and stay gas-tight. The Reuleaux shape is the conceptual seed; the production profile is tuned to the housing.

Reuleaux triangle vs the circle vs an ordinary triangle

	Reuleaux triangle	Circle	Equilateral triangle
Constant width?	Yes	Yes	No
Rolls a load smoothly?	Yes (on free rollers)	Yes	No (jolts)
Centre stays level when rolling?	No (centroid bobs)	Yes	No
Usable as an axle wheel?	No	Yes	No
Perimeter	π·w (Barbier)	π·w	3·side (no single width)
Area	≈ 0.7048 w² (width w)	≈ 0.7854 w² (width w)	≈ 0.4330·side² (width varies)
Corners	3 sharp vertices (120° interior)	None	3 sharp vertices (60°)
Drills a near-square hole?	Yes	No (round)	No

Common misconceptions and pitfalls

• "It's a wheel that isn't round." It rolls a top surface smoothly, but its centre bobs up and down, so it cannot be mounted on a fixed axle. Always distinguish "rolls a load on loose rollers" from "works as a hub-mounted wheel."
• "The square-hole drill makes a perfect square." It leaves four small rounded corners (elliptical arcs) totalling only about 1.2% of the square's area. The hole is about 98.8% of a true square — excellent for mortises, not for a perfectly sharp internal corner.
• "You can chuck the bit in an ordinary drill press." No. The bit's centre must follow a specific non-circular path, so a floating guide template plus an Oldham-style (sliding) coupling is mandatory. A rigid chuck just spins it about its centroid and bores a rounded blob.
• "The Wankel rotor is exactly a Reuleaux triangle." It is Reuleaux-inspired but profiled to its epitrochoidal housing for sealing; the true rotor flanks differ from pure circular arcs.
• "Constant width means constant area swept or constant centre." Constant width is only about the calliper gap. Barbier's theorem fixes the perimeter, but the area is the minimum (Blaschke–Lebesgue) and the centroid is not stationary while rolling.
• "A coin shaped like this won't work in machines built for round coins." The opposite is true: constant width is precisely what lets a slot, calliper, or rolling-gauge read the same diameter at any orientation, so Reuleaux-heptagon coins feed through round-coin mechanisms.