Reuleaux triangle - Printable Version +- MKLab (https://mklab.gr) +-- Forum: [INDEX] (https://mklab.gr/forumdisplay.php?fid=1) +--- Forum: MATHEMATICS (https://mklab.gr/forumdisplay.php?fid=3) +---- Forum: ARTICLES (https://mklab.gr/forumdisplay.php?fid=13) +---- Thread: Reuleaux triangle (/showthread.php?tid=1017)

Reuleaux triangle - mklabgr - 07-09-2026 Reuleaux triangle Summary A Reuleaux triangle is one of geometry’s most fascinating shapes because, despite its rounded appearance, it maintains a constant width in every direction. Created by replacing the sides of an equilateral triangle with circular arcs centered on the opposite vertices, it is the simplest example of a constant-width curve other than a circle. Although the shape is named after 19th-century engineer Franz Reuleaux, it was understood long before his time and appeared in the work of figures such as Leonardo da Vinci and Leonhard Euler.  Its unique geometry allows it to roll smoothly while keeping the same overall width, making it valuable in engineering, mathematics, and design. The Reuleaux triangle also serves as the foundation for a broader family of Reuleaux polygons, which can be constructed from other odd-sided polygons and closely approximate any curve of constant width.  Beyond its elegant mathematical properties, the Reuleaux triangle has numerous practical applications. Its constant width makes it useful in mechanisms, specialized drills that create nearly square holes, rotating machinery, coin design, map projections, architectural patterns, and even road signs. Mathematicians have also studied it extensively because it represents several geometric extremes, including having the smallest possible area among all shapes of the same constant width.  The concept extends into three dimensions through related forms such as the Reuleaux tetrahedron and the Meissner tetrahedron, demonstrating how a simple geometric idea can inspire deeper discoveries. By combining striking symmetry with real-world functionality, the Reuleaux triangle highlights the remarkable ways in which abstract mathematics influences engineering, manufacturing, and modern design. ARTICLE