definition of measure due to Caratheodory and a modification of the definition due to W. Gross. Subsequently, A. P. Morse and Randolph [4] obtained positive results using Gillespie measure. As a result of the work of A. S. Besicovitch and P. A. P. Moran [l] and the author [2], the question was settled in the negative for Hausdorff measure. Furthermore in [2], the author obtained positive results for Favard measure and negative results for Sphere measure. However, the original question asked by Randolph concerning Caratheodory measure remained unanswered. Roughly speaking, Randolph proved that the Caratheodory area does not exceed the product of length by length, while the reverse inequality, despite some lower bounds obtained by the author in [3], yielded to no method of attack. The present paper settles the question for Caratheodory measure in the negative. Briefly summarizing our results, in §3 below a new definition of area in Euclidean 3-space is introduced, based on horizontal projection only. This is in contrast to Caratheodory's definition which involves projection in all directions; however for cylinder sets, these two definitions agree (Theorem 3.8). This result is applied in §6. In §4, a compact subset A of the plane is defined and in §5 it is proved that the Caratheodory length of A is greater than 2.22. Then in §6 it is proved that the Caratheodory area of the cylinder in 3-space of unit height above A is less than 2.172. Since in Euclidean one-dimensional space, Lebesgue length and Caratheodory length agree, the set A provides an example of a set for which the area of A X h is strictly less than the length of A multiplied by A.