Lời giải chi tiết:

\( + \)\(\left\{ O \right\} = AC \cap BD \Rightarrow SO \bot \left( {ABCD} \right)\) 

\( + \)Xét \(\Delta ABC\)có \(\widehat {ABC} = {90^0}\) ta có:

\(A{B^2} + B{C^2} = A{C^2}\) (Định lí Pytago)

\(\begin{array}{l} \Leftrightarrow {2^2} + {2^2} = A{C^2} \Leftrightarrow AC = 2\sqrt 2 \\ \Rightarrow OA = OC = OB = OD = \dfrac{{AC}}{2} = \dfrac{{2\sqrt 2 }}{2} = \sqrt 2 \end{array}\)

\( + \)Xét \(\Delta SOC\) có \(\widehat {SOC} = {90^0}\):

\(S{O^2} + O{C^2} = S{C^2}\)(Định lí Pytago)

\( \Leftrightarrow S{O^2} = {2^2} - {\left( {\sqrt 2 } \right)^2} = 2 \Leftrightarrow SO = \sqrt 2 \)

\( \Rightarrow R = \dfrac{{S{C^2}}}{{2.SO}} = \dfrac{{{2^2}}}{{2\sqrt 2 }} = \dfrac{2}{{\sqrt 2 }} = \sqrt 2 \).

Chọn C