Question Number 211603 by MrGaster last updated on 14/Sep/24
$$\mathrm{1}.\mathrm{Given}\:\mathrm{a}\:\mathrm{regular}\:\mathrm{tetrahedron}\:\boldsymbol{{ABCD}} \\ $$$$\mathrm{with}\:\mathrm{vertices}\:\boldsymbol{{A}}\left(\mathrm{0},\mathrm{0},\mathrm{0}\right)\boldsymbol{{B}}\left(\boldsymbol{{a}},\mathrm{0},\mathrm{0}\right), \\ $$$$\boldsymbol{{C}}\left(\mathrm{0},\boldsymbol{{a}},\mathrm{0}\right),\boldsymbol{\mathrm{and}}\:\boldsymbol{{D}}\left(\mathrm{0},\mathrm{0},\boldsymbol{{a}}\right).\mathrm{Calculate}\:\mathrm{the} \\ $$$$\:\mathrm{volume}\:\boldsymbol{{V}}\:\:\mathrm{and}\:\mathrm{the}\:\mathrm{surface}\:\mathrm{area}\:\boldsymbol{{S}}\:\boldsymbol{\mathrm{of}} \\ $$$$\mathrm{this}\:\mathrm{tetrahedron}. \\ $$
Answered by BHOOPENDRA last updated on 14/Sep/24
Answered by BHOOPENDRA last updated on 15/Sep/24
$${The}\:{volume}\:{of}\:{a}\:{tetrahedron}\:{with}\: \\ $$$${vertices}\:{A},{B},{C},\&{D}\:{is}\:{given}\:{formula} \\ $$$${V}=\frac{\mathrm{1}}{\mathrm{6}}\mid{A}.\left({B}×{C}\right)\mid \\ $$$${A}\left(\mathrm{0},\mathrm{0},\mathrm{0}\right)\:,{B}\left({a},\mathrm{0},\mathrm{0}\right)\:,{C}\left(\mathrm{0},{a},\mathrm{0}\right){D}\left(\mathrm{0},\mathrm{0},{a}\right) \\ $$$$ \\ $$$${V}=\frac{\mathrm{1}}{\mathrm{6}}\begin{vmatrix}{\mathrm{0}\:\:\mathrm{0}\:\:\:\mathrm{0}\:}\\{{a}\:\:\mathrm{0}\:\:\:\mathrm{0}\:\:\:}\\{\mathrm{0}\:\:\:{a}\:\:\:\mathrm{0}}\\{\mathrm{0}\:\:\:\:\mathrm{0}\:\:\:{a}}\end{vmatrix}=\frac{\mathrm{1}}{\mathrm{6}}\mid{a}^{\mathrm{3}} \mid \\ $$$${V}=\frac{\mathrm{1}}{\mathrm{6}}{a}^{\mathrm{3}} \\ $$$${Surface}\:{area}\:=\left(\mathrm{3}×{area}\:{right}\:{angle}\:{triangle}\right)+{equiletral}\:{triangle}\:{area} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{3}×\frac{\mathrm{1}}{\mathrm{2}}{a}^{\mathrm{2}} +\frac{\sqrt{\mathrm{3}}}{\mathrm{4}}\:\left(\sqrt{\mathrm{2}}\:{a}\right)^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\frac{\left(\mathrm{3}+\sqrt{\mathrm{3}}\right){a}^{\mathrm{2}} }{\:\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$
Commented by BHOOPENDRA last updated on 14/Sep/24
$${It}\:{can}\:{be}\:{also}\:{solved}\:{by}\:{using}\: \\ $$$${integration} \\ $$
Answered by mr W last updated on 15/Sep/24
$${V}=\frac{{base}\:{area}×{height}}{\mathrm{3}}=\frac{\mathrm{0}.\mathrm{5}\:{a}^{\mathrm{2}} ×{a}}{\mathrm{3}}=\frac{{a}^{\mathrm{3}} }{\mathrm{6}} \\ $$$${S}=\mathrm{3}×\frac{{a}^{\mathrm{2}} }{\mathrm{2}}+\frac{\sqrt{\mathrm{3}}\left(\sqrt{\mathrm{2}}{a}\right)^{\mathrm{2}} }{\mathrm{4}}=\frac{\left(\mathrm{3}+\sqrt{\mathrm{3}}\right){a}^{\mathrm{2}} }{\mathrm{2}} \\ $$
Commented by BHOOPENDRA last updated on 15/Sep/24
$${Thanks}\:{Mr}.{W}\:{i}\:{have}\:{corrected}\: \\ $$
Commented by mr W last updated on 15/Sep/24