EN|RU Volume 15, No 3, 2008, P. 65-73 UDC 519.178 Ch. Audet, P. Hansen, and F. Messine Ranking small regular polygons by area and by perimeter Abstract: From the pentagon onwards, for each odd number $n$ the area of the regular convex polygon with $n$ sides and unit diameter is greater than the area of the similar polygon with $n + 1$ sides. Moreover, from the heptagon onwards, the difference in areas decreases when $n$ increases. Similar properties hold for the perimeter. A new proof of the Reinhardt’s result is obtained.  Tabl. 1, illustr. 1, bibl. 18. Keywords: polygon, diameter, area, perimeter. Charles Audet 1 Pierre Hansen 2,3 Frederic Messine 4 1. GERAD and Département de Mathématiques et de Génie Industriel, École Polytechnique de Montréal, C.P. 6079, Succ. Centreville, Montréal (Québec), H3C 3A7 Canada  2. GERAD and Département des Méthodes Quantitatives de Gestion, HEC Montréal, 3000 Chemin de la cote Sainte Catherine, Montr´eal H3T 2A7, Canada 3. École des Hautes Études Commerciales de Montréal  C.P. 6079, Succ. Centre-ville, Montréal (Québec), H3C 3A7 Canada 4. Enseeiht-Irit, UMR-CNRS 5828, 2 rue Camichel, 31000 Toulouse, France e-mail: Charles.Audet@gerad.ca, Pierre.Hansen@gerad.ca, Frederic.Messine@n7.fr © Sobolev Institute of Mathematics, 2015