Note on Problem 33

Abstract

Note on Problem 33. By Henry Heaton, Atlantic, Iowa. The result obtained in the published solution of this problem cannot be correct. The area of the regular pentagon is 3.6327X2. The area of each of the infinite number of regular polygons whose apothem is a and number of sides greater than five, is less than 3.6327a2, while that of only two, the square and triangle, is greater. Hence the average area of all regular polygons with apothem a is less than 3.6327a2. Hence the result obtained in the published solution (3.8693a2) is too large. In a similar manner it may be shown that any result larger than al 7 is too large, while it is evi;dent that any result smaller than that is too small.