We investigate the scale-invariant structure of the critical Hardy inequality in a unit ball under the power-type scaling. First we consider the remainder term of the critical Hardy inequality which is characterized by the ratio with or the distance from the “virtual minimizer” for the associated variational problem. We also focus on the scale invariance property of the inequality under power-type scaling and investigate the iterated scaling structure of remainder terms. Finally, we give a relation between the usual scaling enjoyed by the classical Hardy inequality and the power-type scaling via the transformation introduced by Horiuchi and Kumlin. As a by-product, we give a relationship between the Moser sequences and the Talenti functions.