По условию
S_( Δ ABC)=4sqrt(3)
[m]\frac{a^{2}\sqrt{3}}{4}[/m]=4sqrt(3) ⇒ а^2=16 ⇒[blue] a=4[/blue]
V(пирамиды)=(1/3)*S_(осн)*Н
По условию
S_(осн)=[blue]4sqrt(3)[/blue]
V(пирамиды)=8/sqrt(3)
(1/3)*([blue]4sqrt(3)[/blue])*Н=[m]\frac{8}{\sqrt{3}}[/m]
[blue]H=2[/blue]
По теореме Пифагора
SA^2=SO^2+АО^2
SO=H=2
AO=R=[m]\frac{a\sqrt{3}}{3}[/m]=[m]\frac{4\sqrt{3}}{3}[/m]
SA^2=2^2+([m]\frac{4\sqrt{3}}{3}[/m])^2=4+([m]\frac{48}{9}[/m])=
=[m]\frac{84}{9}=\frac{28}{3}[/m]
[m]\sqrt{28}=\sqrt{4\cdot 7}=2\sqrt{7}[/m]
SA=[m]2\cdot \sqrt{\frac{7}{3}}[/m]
О т в е т.[red] [m]2\cdot \sqrt{\frac{7}{3}}[/m][/red]