Editing Reuleaux triangle (section)

=== Other measures ===
By [[Barbier's theorem]] all curves of the same constant width including the Reuleaux triangle have equal [[perimeter]]s. In particular this perimeter equals the perimeter of the circle with the same width, which is <math>\pi s</math>.<ref>{{citation
 | last = Lay | first = Steven R.
 | at = Theorem 11.11, pp. 81–82
 | isbn = 978-0-486-45803-8
 | publisher = Dover
 | title = Convex Sets and Their Applications
 | url = https://books.google.com/books?id=U9eOPjmaH90C&pg=PA81
 | year = 2007}}.</ref><ref>{{citation
 | last = Barbier | first = E.
 | journal = [[Journal de Mathématiques Pures et Appliquées]] | series = 2<sup>e</sup> série
 | language = fr
 | pages = 273–286
 | title = Note sur le problème de l'aiguille et le jeu du joint couvert
 | url = http://sites.mathdoc.fr/JMPA/PDF/JMPA_1860_2_5_A18_0.pdf
 | volume = 5
 | year = 1860}}. See in particular pp. 283–285.</ref><ref name="gardner">{{citation|title=Knots and Borromean Rings, Rep-Tiles, and Eight Queens|volume=4|series=The New Martin Gardner Mathematical Library|first=Martin|last=Gardner|author-link=Martin Gardner|publisher=Cambridge University Press|year=2014|isbn=978-0-521-75613-6|contribution=Chapter 18: Curves of Constant Width|pages=223–245}}.</ref>

The radii of the largest [[inscribed circle]] of a Reuleaux triangle with width ''s'', and of the [[circumscribed circle]] of the same triangle, are
:<math>\displaystyle\left(1-\frac{1}{\sqrt 3}\right)s\approx 0.423s \quad \text{and} \quad \displaystyle\frac{s}{\sqrt 3}\approx 0.577s</math>
respectively; the sum of these radii equals the width of the Reuleaux triangle. More generally, for every curve of constant width, the largest inscribed circle and the smallest circumscribed circle are concentric, and their radii sum to the constant width of the curve.<ref>{{harvtxt|Lay|2007}}, Theorem 11.8, [https://books.google.com/books?id=U9eOPjmaH90C&pg=PA80 pp.&nbsp;80–81].</ref>

{{unsolved|mathematics|How densely can Reuleaux triangles be packed in the plane?}}
The optimal [[packing density]] of the Reuleaux triangle in the plane remains unproven, but is conjectured to be
:<math>\frac{2(\pi-\sqrt 3)}{\sqrt{15}+\sqrt{7}-\sqrt{12}} \approx 0.923, </math>
which is the density of one possible [[double lattice]] packing for these shapes. The best proven upper bound on the packing density is approximately 0.947.<ref>{{citation
 | last1 = Blind | first1 = G.
 | last2 = Blind | first2 = R. | author2-link = Roswitha Blind
 | issue = 2–4
 | journal = Studia Scientiarum Mathematicarum Hungarica
 | language = de
 | mr = 787951
 | pages = 465–469
 | title = Eine Abschätzung für die Dichte der dichtesten Packung mit Reuleaux-Dreiecken
 | volume = 18
 | year = 1983}}. See also {{citation
 | last1 = Blind | first1 = G.
 | last2 = Blind | first2 = R.
 | doi = 10.1007/BF03323256
 | issue = 1–2
 | journal = [[Results in Mathematics]]
 | language = de
 | mr = 880190
 | pages = 1–7
 | title = Reguläre Packungen mit Reuleaux-Dreiecken
 | volume = 11
 | year = 1987| s2cid = 121633860
 }}.</ref> It has also been conjectured, but not proven, that the Reuleaux triangles have the highest packing density of any curve of constant width.<ref>{{citation|arxiv=1504.06733|title=On Curves and Surfaces of Constant Width|year=2015|author-link1=Howard L. Resnikoff|first=Howard L.|last=Resnikoff|bibcode=2015arXiv150406733R}}.</ref>