Câu hỏi 35

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

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

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

\( \Leftrightarrow {a^2} + {a^2} = A{C^2} \Rightarrow AC = a\sqrt 2 \)

\( \Rightarrow OA = OB = OC = OD = \dfrac{{AC}}{2} = a\dfrac{{\sqrt 2 }}{2}\)

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

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

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

\(\begin{array}{l} \Rightarrow R = \dfrac{{S{C^2}}}{{2.SO}} = \dfrac{{{a^2}}}{{2.\dfrac{{\sqrt 2 }}{2}a}} = \dfrac{{\sqrt 2 }}{2}a\\ \Rightarrow S = 4\pi .{R^2} = 4\pi {\left( {\dfrac{{\sqrt 2 }}{2}a} \right)^2} = 2\pi {a^2}\end{array}\)

Chọn C