Significant error in significant figures?

Significant figures are one of the basic starting steps in natural science, even for high school students. The convenient rule-of-thumb on handling significant figures comes from the top constraint that the answer you obtain should not be more precise than the numbers you measured with. Recently, we found a kind of ‘significant error’ in the conventional use of the rule-of-thumb for handling significant figures. Here, we show that the application of the convenient rule-of-thumb can easily and widely violate the top constraint of significant figures when calculating the areas of squares and the volumes of cubes. We believe that this ‘significant error’ should be taught in the elementary physics courses for most freshmen in the science/engineering colleges.