Question Number 19955 by ajfour last updated on 19/Aug/17
Commented by ajfour last updated on 19/Aug/17
$${Find}\:{the}\:{volume}\:{of}\:{the}\:{regular} \\ $$$${tetrahedron}\:{in}\:{terms}\:{of}\:\boldsymbol{{r}}\:. \\ $$
Commented by ajfour last updated on 19/Aug/17
Answered by dioph last updated on 19/Aug/17
$$\left(\mathrm{2}{r}\right)^{\mathrm{2}} \:=\:{l}^{\mathrm{2}} \:+\:\left(\frac{{l}}{\:\sqrt{\mathrm{2}}}\right)^{\mathrm{2}} \\ $$$$\mathrm{4}{r}^{\mathrm{2}} \:=\:{l}^{\mathrm{2}} \:+\:\frac{{l}^{\mathrm{2}} }{\mathrm{2}} \\ $$$${r}^{\mathrm{2}} \:=\:\frac{\mathrm{3}{l}^{\mathrm{2}} }{\mathrm{8}}\:\therefore\:{l}\:=\:\frac{\mathrm{2}{r}\sqrt{\mathrm{2}}}{\:\sqrt{\mathrm{3}}} \\ $$$${h}^{\mathrm{2}} \:+\:\left(\frac{{l}\sqrt{\mathrm{3}}}{\mathrm{3}}\right)^{\mathrm{2}} \:=\:{l}^{\mathrm{2}} \\ $$$${h}^{\mathrm{2}} \:=\:{l}^{\mathrm{2}} \:−\:\frac{{l}^{\mathrm{2}} }{\mathrm{3}} \\ $$$${h}\:=\:\frac{{l}\sqrt{\mathrm{2}}}{\:\sqrt{\mathrm{3}}} \\ $$$${V}\:=\:\frac{\mathrm{1}}{\mathrm{3}}\left(\frac{{l}^{\mathrm{2}} \sqrt{\mathrm{3}}}{\mathrm{4}}\:×\:{h}\right)\:=\:\frac{{l}^{\mathrm{3}} \sqrt{\mathrm{2}}}{\mathrm{12}} \\ $$$$\Rightarrow\:{V}\:=\:\frac{\mathrm{8}{r}^{\mathrm{3}} ×\mathrm{2}\sqrt{\mathrm{2}}}{\mathrm{3}\sqrt{\mathrm{3}}}×\frac{\sqrt{\mathrm{2}}}{\mathrm{12}}\:=\:\frac{\mathrm{32}{r}^{\mathrm{3}} }{\mathrm{36}\sqrt{\mathrm{3}}}\:=\:\frac{\mathrm{8}{r}^{\mathrm{3}} }{\mathrm{9}\sqrt{\mathrm{3}}} \\ $$
Commented by ajfour last updated on 19/Aug/17
$${Excellent}\:{Sir},\:{thanks}\:{a}\:{lot}. \\ $$$${but}\:{the}\:{first}\:{line},\:\frac{{l}}{\:\sqrt{\mathrm{2}}}\:;\:{i}\:{did}\:{not} \\ $$$${understand}.. \\ $$
Commented by dioph last updated on 19/Aug/17
Commented by dioph last updated on 19/Aug/17
$$\mathrm{My}\:\mathrm{poor}\:\mathrm{attempt}\:\mathrm{on}\:\mathrm{drawing}\:\mathrm{a} \\ $$$$\mathrm{tetrahedron}\:\mathrm{in}\:\mathrm{a}\:\mathrm{way}\:\mathrm{that}\:\mathrm{its}\:\mathrm{edges} \\ $$$$\mathrm{are}\:\mathrm{the}\:\mathrm{diagonals}\:\mathrm{of}\:\mathrm{the}\:\mathrm{faces}\:\mathrm{of} \\ $$$$\mathrm{a}\:\mathrm{cube}. \\ $$$$\mathrm{The}\:\mathrm{main}\:\mathrm{diagonal}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cube}\:\mathrm{is} \\ $$$$\mathrm{2}{r},\:\mathrm{hence}\:\left(\mathrm{2}{r}\right)^{\mathrm{2}} \:=\:{l}^{\mathrm{2}} \:+\:\left(\frac{{l}}{\:\sqrt{\mathrm{2}}}\right)^{\mathrm{2}} \\ $$
Commented by ajfour last updated on 19/Aug/17
$${thanks}\:{sir},\:{all}\:{the}\:{more}. \\ $$