Solution: Sphere surface area: $ 4\pi r^2 $. Circle area: $ \pi (\sqrt2r)^2 = 2\pi r^2 $. Setting equal: $ 4\pi r^2 = 2\pi r^2 $. This implies $ 4 = 2 $, a contradiction. Thus, no solution exists unless the circle’s radius is adjusted. However, if the problem states equivalence, the only possibility is $ r = 0 $, which is trivial. Rechecking the question reveals a misstatement; assuming the circle’s radius is $ R $, then $ 4\pi r^2 = \pi R^2 \Rightarrow R = 2r $. The original question’s setup is inconsistent, but if forced, $ r = \fracR2 $, so $ \boxedr = \dfracR2 $. - Appfinity Technologies