Barbier's theorem

These Reuleaux polygons have constant width, and all have the same width; therefore by Barbier's theorem they also have equal perimeters.

In geometry, Barbier's theorem states that every curve of constant width has perimeter π times its width, regardless of its precise shape.[1] This theorem was first published by Joseph-Émile Barbier in 1860.[2]

  1. ^ Lay, Steven R. (2007), Convex Sets and Their Applications, Dover, Theorem 11.11, pp. 81–82, ISBN 9780486458038.
  2. ^ Barbier, E. (1860), "Note sur le problème de l'aiguille et le jeu du joint couvert" (PDF), Journal de mathématiques pures et appliquées, 2e série (in French), 5: 273–286, archived from the original (PDF) on 2017-04-20. See in particular pp. 283–285.