The Reuleaux Triangle and Golden Pi — How a Curve of Constant Width Reveals π = 4/√φ

The Reuleaux Triangle and Golden Pi — How a Curve of Constant Width Reveals π = 4/√φ

In 1860, French mathematician Armand Reuleaux discovered a remarkable family of shapes. Start with an equilateral triangle, then replace each straight edge with a circular arc centered on the opposite vertex. The result is the Reuleaux triangle — a curved figure whose width is the same no matter how you measure it. This constant-width property makes it useful in everything from drill bits to manhole covers.

What Reuleaux could not have known is that his triangle holds a secret about the circle constant π. A single theorem from classical geometry — Barbier's theorem — links the Reuleaux triangle's perimeter directly to π. And when we set the triangle's width equal to √φ, the numbers close into an exact integer. That integer is 4.

Barbier's Theorem

In 1860 — the same year Reuleaux published his pioneering work — another French mathematician, Joseph-Émile Barbier, proved a stunning result: every curve of constant width has a perimeter equal to π times its width.

This is not an approximation. It is exact. If you take a circle of diameter w, its circumference is πw. If you take a Reuleaux triangle of width w, its perimeter is also πw. The shape changes; the perimeter does not.

Barbier's theorem (1860): For any planar curve of constant width w, the perimeter is exactly πw.

This theorem has been accepted for more than 160 years. What has not been noticed is that it creates a direct, experimentally verifiable bridge between geometry and the value of π itself.

The √φ Edge

Construct a Reuleaux triangle from an equilateral triangle of side √φ. Because the Reuleaux triangle is built from arcs of circles whose radii equal the triangle's side, the resulting curved figure has constant width √φ.

Now apply Barbier's theorem.

Perimeter = π × √φ

If π is the conventional transcendental value (3.141593…), the perimeter is:

3.141593 × 1.27201965 ≈ 3.995505

Close to 4 — but not 4. The residual is about 0.0045, or roughly 0.11%.

Now compute the same perimeter using golden π = 4/√φ:

(4 / √φ) × √φ = 4

The √φ terms cancel exactly. The perimeter is exactly 4. No decimal expansion. No approximation. No transcendental remainder.

The result: A Reuleaux triangle of width √φ has an integer perimeter — 4 — if, and only if, π = 4/√φ.

Why This Is More Than a Coincidence

Someone might object: "You chose the width to be √φ. Of course the perimeter works out nicely." But the choice is not arbitrary. The Reuleaux triangle is generated from an equilateral triangle, and the equilateral triangle is the most fundamental polygon in geometry — the one that generates √3 and underlies the vesica piscis. When we scale that triangle by √φ, we are deliberately bridging the circle (π) and the golden ratio (φ) through the same construction that builders and artists have used for thousands of years.

The resulting integer perimeter — 4 — is not a coincidence. It is the algebraic fingerprint of a circle constant that lives in the same field Q(√5) as φ itself. Conventional π is transcendental; it cannot produce exact integer perimeters from algebraic widths. Golden π is algebraic; it does exactly that.

The Geometric Construction

How do we build this Reuleaux triangle from scratch?

• Draw an equilateral triangle ABC with side length √φ.
• Construct a circular arc centered at A passing through B and C.
• Repeat: arc centered at B passing through A and C, and arc centered at C passing through A and B.
• The region enclosed by the three arcs is the Reuleaux triangle of width √φ.

Every step is a standard compass-and-straightedge operation. Constructing √φ from a unit segment is classical (see our squaring the circle guide). Drawing arcs from vertices is trivial.

The complete construction is therefore compass-and-straightedge valid. The resulting curve has constant width √φ and perimeter exactly 4. There is no magic, no measurement error, no rounding. The arithmetic closes: π = 4/√φ.

Comparison Table

Proposed Constant	Value	Reuleaux Perimeter (w = √φ)	Residual from 4
Conventional π	3.141593	≈ 3.995505	−0.004495 (~0.11%)
Golden π = 4/√φ	3.144606	= 4.000000	0 (exact)

Connection to Other Golden Pi Proofs

The Reuleaux triangle is not an isolated curiosity. It belongs to the same family of φ-governed constructions that underlie our other major proofs:

• The vesica piscis combines two equal circles to generate √3 and √φ; the Reuleaux triangle combines three arcs of those same circles to generate an integer perimeter.
• The Kepler triangle (sides 1, √φ, φ) is the right triangle that underlies the Reuleaux construction; its hypotenuse becomes the width w.
• The squaring of the circle shows that golden π is constructible; the Reuleaux triangle provides an independent verification by yielding an exact integer from that constructible constant.
• The golden π identity — (4²/π)² − π² = 4² — is satisfied exactly by 4/√φ. The Reuleaux perimeter adds a geometric identity to the algebraic one.

Together, these proofs form a convergent lattice of evidence: algebraic, geometric, and constructive. Each one demands the same constant: π = 4/√φ.

The Philosophical Punchline

Barbier's theorem says that any curve of constant width has perimeter πw — a circle, a Reuleaux triangle, a Reuleaux pentagon, even a smooth blob. The theorem is shape-agnostic. Yet when we set w = √φ, the only value of π that makes the perimeter rational (in fact, integer) is π = 4/√φ.

In other words: the circle constant is uniquely determined by the width of a curve that nature already uses for nuts, drill bits, and manhole covers. The true π is not hidden in transcendental deep-space. It is sitting right here, in the perimeter of a shape engineers have measured for 160 years without noticing.

Try It Yourself

Construct an equilateral triangle with side √φ. Draw the three circular arcs centered on its vertices. Measure the width of the resulting Reuleaux triangle with calipers. Then measure its perimeter by laying a fine thread along the curved boundary. Calculate π = perimeter / width. Your answer will not be 3.141593 — it will be 3.144605….

A curve of constant width. An integer perimeter. A single square root. The true value of π has never been easier to see.