page_title: Reuleaux triangle
section_title: Reuleaux polygons
hierarchical_section_title: Reuleaux triangle / Generalizations / Reuleaux polygons
caption_attribution_description: English: The front of a fifty fil coin from the United Arab Emirates
context_page_description: A Reuleaux triangle is a shape formed from the intersection of three circular disks, each having its center on the boundary of the other two. Its boundary is a curve of constant width, the simplest and best known such curve other than the circle itself. Constant width means that the separation of every two parallel supporting lines is the same, independent of their orientation. Because all its diameters are the same, the Reuleaux triangle is one answer to the question "Other than a circle, what shape can a manhole cover be made so that it cannot fall down through the hole?"
  Reuleaux triangles have also been called spherical triangles, but that term more properly refers to triangles on the curved surface of a sphere.
  They are named after Franz Reuleaux, a 19th-century German engineer who pioneered the study of machines for translating one type of motion into another, and who used Reuleaux triangles in his designs. However, these shapes were known before his time, for instance by the designers of Gothic church windows, by Leonardo da Vinci, who used it for a map projection, and by Leonhard Euler in his study of constant-width shapes.
context_section_description: The Reuleaux triangle can be generalized to regular polygons with an odd number of sides, yielding a Reuleaux polygon. These are the only curves of constant width whose boundaries are formed by finitely many circular arcs of equal length.
  The constant width of these shapes allows their use as coins that can be used in coin-operated machines. For instance, the United Kingdom has made 20-pence and 50-pence coins in the shape of a Reuleaux heptagon. The Canadian Loonie dollar coin uses another Reuleaux polygon with 11 sides.
  Similar methods can be used to enclose an arbitrary simple polygon within a curve of constant width, whose width equals the diameter of the given polygon. The resulting shape consists of circular arcs (at most as many as sides of the polygon), can be constructed algorithmically in linear time, and can be drawn with compass and straightedge. Although the regular-polygon based Reuleaux polygons all have an odd number of circular-arc sides, it is possible to construct constant-width shapes based on irregular polygons that have an even number of sides.
caption: United Arab Emirates 50 fils coin, a Reuleaux heptagon in form.
exif: {"Image ImageWidth": "1618", "Image Model": "ONEPLUS A5000", "Image ImageLength": "1618", "Image Make": "OnePlus", "Image ExifOffset": "152", "Image Orientation": "0", "Image DateTime": "2018:05:27 00:17:06", "GPS GPSLatitude": "[24, 26, 7782/625]", "GPS GPSAltitude": "0", "GPS GPSLatitudeRef": "N", "GPS GPSAltitudeRef": "0", "GPS GPSLongitudeRef": "E", "GPS GPSTimeStamp": "[20, 17, 5]", "GPS GPSLongitude": "[54, 27, 157391/10000]", "GPS GPSDate": "2018:05:26", "Image GPSInfo": "266", "EXIF WhiteBalance": "Auto", "EXIF ISOSpeedRatings": "125", "EXIF FocalLength": "4103/1000", "EXIF ExposureTime": "3/2000", "EXIF Flash": "Flash did not fire, compulsory flash mode", "EXIF LightSource": "Unknown", "EXIF FNumber": "17/10"}