Răspuns
apotema = inaltimea unui triunghi echilateral = [tex]\frac{l\sqrt{3} }{2}[/tex]
⇒ [tex]\frac{l\sqrt{3} }{2} = 2\sqrt{3}[/tex] ⇒ [tex]l\sqrt{3} = 2 * 2\sqrt{3}[/tex]
l = 4
Aria bazei = [tex]\frac{l^{2}\sqrt{3}}{4} = \frac{16\sqrt{3}}{4} = 4\sqrt{3}[/tex]
Apotema bazei = [tex]\frac{1}{3} * h = \frac{1}{3} * 2\sqrt{3} = \frac{2\sqrt{3}}{3}[/tex]
Aplicam teorema lui Pitagora sa aflam inaltimea piramidei
[tex]h^{2} = (2\sqrt{3}) ^{2} - (\frac{2\sqrt{3} }{3}) ^{2} \\h^{2} = 12 - \frac{12}{9} \\\h = \sqrt{\frac{108}{9} } \\h = \frac{4\sqrt{6} }{3}[/tex]
Volumul = [tex]\frac{Ab*h}{3} = \frac{4\sqrt{3}*\frac{4\sqrt{3}}{6}}{3} = \frac{16\sqrt{2} }{3} cm^{3}[/tex]
Explicație pas cu pas:
Tetraedrul regulat are toate fetele triunghiuri echilaterale.
