The circle is known to produce the maximum Area to Perimeter ratio, which is equal to a constant D/4, D being the circle diameter. But, this constant is not unique to the circle and is shared with other closed spaces inscribing the circle. This maximal relationship is deduced through the rationalist paradigm of geometry and mathematics. By using a constructivist design paradigm, shapes are found which paradoxically exceed the maximum ratio of D/4. Beside the practical applications of these geometric paradoxical shapes, the paradox itself is explained by identifying two different approaches to contructive problem solving. One of these shapes is used as a model to draw a direct comparison between the different approaches and particularly between objective rationalism and subjective constructivism, in the form of constructive design. The comparison provides some insights, differences and consolidation of theories of Rationalism, Constructivism, Holism and Godel’s Incompleteness Theorems. Further thoughts are expressed to demonstrate the value of constructs in creating wholes with different properties; to support the use of heuristics; and to discuss some outcomes and implications of using different paradigms.

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