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Reuleaux Triangle — Definition, Formula & Examples

A Reuleaux triangle is a curved shape built from three circular arcs, each centered at one vertex of an equilateral triangle and passing through the other two vertices. It has the special property of constant width — the distance between any two parallel supporting lines is always the same.

Given an equilateral triangle with side length ss, the Reuleaux triangle is the intersection of three closed disks of radius ss, each centered at a vertex of the triangle. It is the simplest non-circular curve of constant width ss.

Key Formula

A=12(π−3) s2A = \frac{1}{2}(\pi - \sqrt{3})\,s^2
Where:
  • AA = Area of the Reuleaux triangle
  • ss = Side length of the underlying equilateral triangle (also the constant width)

How It Works

To construct a Reuleaux triangle, start with an equilateral triangle of side length ss. From each vertex, draw a circular arc of radius ss connecting the other two vertices. The region enclosed by these three arcs is the Reuleaux triangle. Because every arc has the same radius ss, any pair of parallel tangent lines touching opposite sides will always be exactly ss apart. This constant-width property means a Reuleaux triangle can rotate freely inside a square of side length ss, staying in contact with all four sides — a principle used in drill bits that cut nearly square holes.

Worked Example

Problem: Find the perimeter and area of a Reuleaux triangle with constant width 6 cm.
Identify the arcs: The shape consists of three circular arcs, each subtending an angle of 60° (i.e., π/3 radians) on a circle of radius s = 6 cm.
Compute the perimeter: Each arc has length rθ = 6 · (π/3) = 2π. Three arcs give the total perimeter.
P=3×2π=6π≈18.85 cmP = 3 \times 2\pi = 6\pi \approx 18.85 \text{ cm}
Compute the area: Apply the area formula with s = 6.
A=12(π−3)(6)2=18(π−3)≈25.37 cm2A = \frac{1}{2}(\pi - \sqrt{3})(6)^2 = 18(\pi - \sqrt{3}) \approx 25.37 \text{ cm}^2
Answer: The perimeter is 6π≈18.856\pi \approx 18.85 cm and the area is 18(π−3)≈25.3718(\pi - \sqrt{3}) \approx 25.37 cm².

Why It Matters

Reuleaux triangles appear in engineering: Wankel rotary engines use a rotor shaped like a Reuleaux triangle, and specialized drill bits exploit the constant-width property to mill nearly square holes. In mathematics, they provide a key example in the study of curves of constant width and help prove Barbier's theorem, which states that every curve of constant width ww has perimeter πw\pi w.

Common Mistakes

Mistake: Assuming the Reuleaux triangle has straight sides like a regular triangle.
Correction: Every side is a circular arc, not a line segment. The vertices are the only sharp points; the edges curve outward.