Ranking small regular polygons by area and by perimeter

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.