# Reuleaux triangle
The Reuleaux triangle is a curve of constant width formed from an equilateral triangle by replacing each side with a circular arc centered at the opposite vertex, so that every pair of parallel supporting lines touches it at the same distance apart. It is the simplest non-circular curve of constant width, and among all curves of a given constant width it encloses the smallest area.
| Key fact | Detail |
|---|---|
| Construction | Three circular arcs of radius equal to the side of an equilateral triangle, each centered at one vertex and joining the other two vertices[1](https://mathworld.wolfram.com/ReuleauxTriangle.html) |
| Width | Equal to the side length of the generating equilateral triangle[2](https://encyclopediaofmath.org/wiki/Constant_width%2C_curve_of) |
| Area | \( A = \tfrac{1}{2}(\pi - \sqrt{3})\,r^{2} \) for arc radius \( r \)[1](https://mathworld.wolfram.com/ReuleauxTriangle.html) |
| Perimeter | \( \pi w \) for width \( w \), the same as a circle of diameter \( w \) (Barbier's theorem)[2](https://encyclopediaofmath.org/wiki/Constant_width%2C_curve_of) |
| Extremal property | Smallest area of any curve of given constant width (Blaschke–Lebesgue theorem); the circle has the largest[3](https://arxiv.org/abs/math/0009137) • [4](https://www.sciencedirect.com/science/article/abs/pii/S0723086924000847) |
| Rotation in a square | Covers about 98.77% of the square's area; the basis of the Harry Watts square drill bit[1](https://mathworld.wolfram.com/ReuleauxTriangle.html) |
| Namesake | Franz Reuleaux (1829–1905), German professor of machine theory, a founder of modern kinematics[5](https://library.ethz.ch/sammlungen-und-archive/kurzportraets/franz-reuleaux--1829-1905-.html) |
## Definition and construction
A curve of constant width is a plane convex curve for which the distance between any two parallel supporting lines is the same, whatever the direction; this distance is the width[2](https://encyclopediaofmath.org/wiki/Constant_width%2C_curve_of). The circle is the familiar example, but there are infinitely many others, generally non-smooth[2](https://encyclopediaofmath.org/wiki/Constant_width%2C_curve_of).
To build the Reuleaux triangle, take an equilateral triangle of side \( r \) and draw three circles of radius \( r \), each centered at one vertex. The intersection of the three disks is the Reuleaux triangle: its boundary consists of three arcs, each joining two vertices and centered at the third[1](https://mathworld.wolfram.com/ReuleauxTriangle.html) • [4](https://www.sciencedirect.com/science/article/abs/pii/S0723086924000847). The construction needs only a compass, since each arc is drawn from a vertex already marked[1](https://mathworld.wolfram.com/ReuleauxTriangle.html).
The constant-width property follows from a single observation: for any pair of parallel supporting lines, one touches a vertex of the equilateral triangle and the other touches a point on the circle of radius \( r \) centered at that vertex, so the distance between the lines is always \( r \)[4](https://www.sciencedirect.com/science/article/abs/pii/S0723086924000847).
## Area and perimeter
The area decomposes into the central equilateral triangle plus three circular segments. Each segment has opening angle \( \theta = \pi/3 \), so the total area is
\[ A = 3 A_{s} + A_{t} = \tfrac{1}{2}(\pi - \sqrt{3})\,r^{2} \]
where \( A_{s} = \tfrac{1}{2}r^{2}(\theta - \sin\theta) \) is the area of one segment and \( A_{t} = \tfrac{\sqrt{3}}{4}r^{2} \) is the area of the central triangle[1](https://mathworld.wolfram.com/ReuleauxTriangle.html). Numerically this is about 0.7048 \( r^{2} \), against \( \pi r^{2}/4 \approx 0.7854\,r^{2} \) for the circle of the same width.
By Barbier's theorem, every curve of constant width \( w \) has perimeter \( \pi w \), the same as the circle of diameter \( w \)[2](https://encyclopediaofmath.org/wiki/Constant_width%2C_curve_of). The Reuleaux triangle therefore has the same perimeter as the equal-width circle while enclosing less area, which is why it, not the circle, minimizes area at fixed width.
## The Blaschke–Lebesgue theorem
The theorem, proved independently by [Wilhelm Blaschke](https://www.edgepedia.org/wilhelm-blaschke) and [Henri Lebesgue](https://www.edgepedia.org/henri-lebesgue), states that among all planar convex domains of a given constant width, the Reuleaux triangle has minimal area[3](https://arxiv.org/abs/math/0009137) • [4](https://www.sciencedirect.com/science/article/abs/pii/S0723086924000847). Harrell's 2000 direct proof analyzes the underlying variational problem and shows uniqueness up to rigid motions, without assuming in advance which shape minimizes[3](https://arxiv.org/abs/math/0009137). The result has been extended to the sphere and to hyperbolic space, where Reuleaux triangles likewise have minimal area among convex domains of constant width[6](https://arxiv.org/abs/2203.16636). In three dimensions the corresponding question remains open: the convex body of given constant width and minimal volume is unknown[3](https://arxiv.org/abs/math/0009137).
## History and the name
The shape is older than its name. It appears in Gothic church window tracery, including windows of the Sagrada Familia, and [Leonardo da Vinci](https://www.edgepedia.org/leonardo-da-vinci) used it in his octant projection[7](https://elpais.com/ciencia/el-juego-de-la-ciencia/2025-10-24/el-triangulo-de-reuleaux.html). Euler analyzed curves of constant width in 1771[8](https://light.artwaste.land/strata/reuleaux-constant-width/).
The name honors Franz Reuleaux (1829–1905), born in Eschweiler, professor of machine theory who taught at the Eidgenössisches Polytechnikum in Zurich from 1856 and then held a chair at the Berlin Trade Academy from 1864 to 1896, serving also as its rector[5](https://library.ethz.ch/sammlungen-und-archive/kurzportraets/franz-reuleaux--1829-1905-.html). His *Lehrbuch der theoretischen Kinematik*, published in [Braunschweig](https://www.edgepedia.org/braunschweig) in 1875 and quickly translated into French, English, and Italian, reduced machine design to mathematical principles and made him a founder of modern kinematics[5](https://library.ethz.ch/sammlungen-und-archive/kurzportraets/franz-reuleaux--1829-1905-.html) • [9](https://ecommons.cornell.edu/entities/publication/51459559-1cc3-4142-b201-6e38804c46a9). His kinematic study and mechanical applications of the curve gave it his name[7](https://elpais.com/ciencia/el-juego-de-la-ciencia/2025-10-24/el-triangulo-de-reuleaux.html).
## Rotation in a square and the square drill
Because a Reuleaux triangle of width \( w \) fits inside a square of side \( w \) at every orientation, it can rotate within that square while keeping all four sides in contact[1](https://mathworld.wolfram.com/ReuleauxTriangle.html). As it turns, its corners sweep out nearly the whole square: the covered fraction is
\[ A_{\text{covered}} = 2\sqrt{3} + \tfrac{1}{6}\pi - 3 = 0.9877003907\ldots \]
The uncovered part is a small region at each corner with a rounded, elliptical boundary[1](https://mathworld.wolfram.com/ReuleauxTriangle.html).
The geometric centroid does not stay fixed during this rotation, nor does it move on a circle; its path is a curve made of four arcs of ellipses[1](https://mathworld.wolfram.com/ReuleauxTriangle.html). This wandering center is why a Reuleaux-triangle wheel would bump its rider up and down at every turn[10](https://lumaths.com/concepts/triangle-reuleaux), and it is also the mechanical fact exploited in drilling: the bit must be mounted in a special guide that lets its center wander.
The Harry Watts square drill bit, patented in 1914, uses a Reuleaux triangle rotated inside a square guide to cut a hole that is nearly square, with slightly rounded corners[8](https://light.artwaste.land/strata/reuleaux-constant-width/). Practical bits are made concave in three spots to clear shavings and permit corner cutting[11](https://www.mikesenese.com/DOIT/2011/10/drilling-square-holes-with-a-reuleaux-triangle/). A related constant-width shape based on an isosceles right triangle can produce an exact square hole[12](https://www-dev.ics.uci.edu/~eppstein/junkyard/reuleaux.html).
## Reuleaux polygons and other relatives
The same construction applied to any regular polygon with an odd number of sides, drawing an arc from each vertex through the two opposite vertices, produces a Reuleaux polygon, also of constant width[7](https://elpais.com/ciencia/el-juego-de-la-ciencia/2025-10-24/el-triangulo-de-reuleaux.html). The constant-width condition requires an odd number of sides[4](https://www.sciencedirect.com/science/article/abs/pii/S0723086924000847). Among all curves of a given constant width, the circle has the largest area and the Reuleaux triangle the smallest, so these two shapes bracket the whole family[4](https://www.sciencedirect.com/science/article/abs/pii/S0723086924000847). Constant-width curves can be closely approximated by Reuleaux polygons, a fact used in proofs of the Blaschke–Lebesgue theorem[4](https://www.sciencedirect.com/science/article/abs/pii/S0723086924000847).
## Applications
The best-known application is the Wankel rotary engine, in which a triangular rotor turns inside an oval housing; the rotor's apices follow the inner periphery of the casing while staying in contact with the gear on the eccentric output shaft[13](https://de.mazda-press.com/api/assets/download/861958d3-4963-4bec-940e-499855ee6992_Default?isDownload=false).
In robotics, a Reuleaux-triangle cam-follower mechanism has been used to generate a bipedal robot's foot trajectory with a straight-line support phase and a constant body height, minimizing the number of active degrees of freedom[14](https://www.mdpi.com/2218-6581/10/4/114). The shape also appears in guitar picks, pencils, and architectural glazing based on three-fold symmetry[15](https://doi.org/10.1007/s00004-015-0253-9).
## References
1. [Weisstein, Eric W. "Reuleaux Triangle." MathWorld, Wolfram Research.](https://mathworld.wolfram.com/ReuleauxTriangle.html)
2. [Sokolov, D. D. "Constant width, curve of." Encyclopedia of Mathematics (European Mathematical Society).](https://encyclopediaofmath.org/wiki/Constant_width%2C_curve_of)
3. [Harrell, Evans M. (2000). "A direct proof of a theorem of Blaschke and Lebesgue." arXiv:math/0009137.](https://arxiv.org/abs/math/0009137)
4. ["The Blaschke–Lebesgue theorem revisited." Expositiones Mathematicae (ScienceDirect).](https://www.sciencedirect.com/science/article/abs/pii/S0723086924000847)
5. ["Franz Reuleaux (1829–1905)." ETH-Bibliothek, ETH Zürich.](https://library.ethz.ch/sammlungen-und-archive/kurzportraets/franz-reuleaux--1829-1905-.html)
6. ["Convex bodies of constant width in spaces of constant curvature and the extremal area of Reuleaux triangles." arXiv:2203.16636.](https://arxiv.org/abs/2203.16636)
7. ["El triángulo de Reuleaux." El País, El juego de la ciencia.](https://elpais.com/ciencia/el-juego-de-la-ciencia/2025-10-24/el-triangulo-de-reuleaux.html)
8. [curves of constant width — shapes that roll like a circle](https://light.artwaste.land/strata/reuleaux-constant-width/)
9. [Reuleaux, Franz (1875). *Lehrbuch der Kinematik*, V.1. Braunschweig: F. Vieweg und sohn. Cornell eCommons.](https://ecommons.cornell.edu/entities/publication/51459559-1cc3-4142-b201-6e38804c46a9)
10. ["Le triangle de Reuleaux." Lumaths.](https://lumaths.com/concepts/triangle-reuleaux)
11. ["Drilling Square Holes with a Reuleaux Triangle." Mike Senese, DO IT.](https://www.mikesenese.com/DOIT/2011/10/drilling-square-holes-with-a-reuleaux-triangle/)
12. ["The Geometry Junkyard: Reuleaux Triangles." David Eppstein, UC Irvine.](https://www-dev.ics.uci.edu/~eppstein/junkyard/reuleaux.html)
13. ["Rotary Engine 2007E." Mazda Press archive.](https://de.mazda-press.com/api/assets/download/861958d3-4963-4bec-940e-499855ee6992_Default?isDownload=false)
14. ["Reuleaux Triangle–Based Two Degrees of Freedom Bipedal Robot." Robotics 10(4):114, MDPI.](https://www.mdpi.com/2218-6581/10/4/114)
15. ["An Introduction to the Vesica Piscis, the Reuleaux Triangle and Related Geometric Constructions in Modern Architecture." Nexus Network Journal, Springer.](https://doi.org/10.1007/s00004-015-0253-9)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Geometry and topology › Metric, convex, and discrete geometry*
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