A criterion is proposed first to quantitatively check the interpolation precision of quadrilateral elements. With this criterion, the necessary geometry constraints are derived for 8-node (Q8) element to exactly attain quadratic interpolation precision. The study is next to focus on improvement of the Q8 element with a bilinear configuration and the Q8 element is subsequently presented by slightly modifying the Q8 element. Theoretical analysis is conducted as well to confirm the interpolation precision predicted in the Q8 element. Through typical numerical examples, it is concluded that of all the tested elements the second-order integrated Q8 element is the most robust one for all cases.