d)
[tex]|OD|=\sqrt{3} , |SD|=3\sqrt{3}[/tex]
[tex]|OD|=\frac{1}{3} |CD|\\|CD|=3*|OD|\\|CD|=3*\sqrt{3} =3\sqrt{3}[/tex]
[tex]|CO|=\frac{2}{3} |CD|\\|CO|=\frac{2}{3} *3\sqrt{3} =2\sqrt{3}[/tex]
[tex]|CD|=\frac{|AB|\sqrt{3} }{2} \\|AB|=\frac{2*|CD|}{\sqrt{3} } \\|AB|=\frac{2*3\sqrt{3} }{\sqrt{3} } =6[/tex]
[tex]|SO|^{2} +|OD|^{2} =|SD|^{2} \\|SO|^{2}+(\sqrt{3}) ^{2} =(3\sqrt{3}) ^{2} \\|SO|^{2}+3=27\\|SO|^{2}=24\\|SO|=\sqrt{24} =2\sqrt{6}[/tex]
[tex]S_{p} =\frac{1}{2} *|AB|*|CD|\\S_{p} =\frac{1}{2} *6*3\sqrt{3} =9\sqrt{3}[/tex]
[tex]V=\frac{1}{3}* S_{p} *|SO|\\V=\frac{1}{3} *9\sqrt{3} *2\sqrt{6} =6\sqrt{18} =6*3\sqrt{2} =18\sqrt{2}[/tex]
[tex]S_{b} =3*\frac{1}{2} *|AB|*|SD|\\S_{b} =3*\frac{1}{2} *6*3\sqrt{3} =27\sqrt{3}[/tex]
