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An implementation of a radix-4 approximate squaring circuit is described employing a new operand dual recoding technique. Approximate squaring circuits have numerous applications including use in computer graphics, digital radio modules, implementation of division and function approximation in ALU circuits. The theory of operation of the circuit is described including radix-4 operand dual recoding. Our recoding yields non negative partial squares and other features which simplify the design of the approximate squaring circuit. Results of the implementation in terms of delay, power, and area in both 130 nm and 90 nm technologies are presented and analyzed. The results show the circuit is power, area and performance efficient, yielding reduction factors by three or more when compared to a truncated multiplication approach using state-of-the-art logic synthesis tools. The radix-4 squaring circuit is also shown to be more efficient than a radix-2 state-of-the-art binary squaring circuit.