Question Number 203319 by ajfour last updated on 16/Jan/24
Commented by mr W last updated on 16/Jan/24
$$\mathrm{4}\:{equal}\:{spheres}\:{inside}\:{a}\:{regular} \\ $$$${tetrahedron}\:{with}\:{edge}\:{length}\:{a}? \\ $$
Commented by ajfour last updated on 16/Jan/24
$${Well}\:{If}\:{each}\:{equal}\:{radii}\:{sphere}\:{touch}\: \\ $$$${then}\:{distance}\:{from}\:{gap}\:{centre}\:{to}\:{sphere} \\ $$$${center}\:{is}\:{how}\:{many}\:{times}\:{r}\:? \\ $$$${I}\:{think}\:{its}\:{simple}\:{r}=\frac{{a}}{\:\sqrt{\mathrm{3}}} \\ $$
Commented by mr W last updated on 17/Jan/24
$${distance}\:{from}\:{center}\:{of}\:{sphere}\:{to}\: \\ $$$${center}\:{of}\:{sphere}\:=\mathrm{2}{r} \\ $$$${distance}\:{from}\:{centroid}\:{to}\:{center}\:{of} \\ $$$${sphere}\:={a}=\frac{\sqrt{\mathrm{6}}\:\left(\mathrm{2}{r}\right)}{\mathrm{4}}=\frac{\sqrt{\mathrm{6}}{r}}{\mathrm{2}} \\ $$$$\Rightarrow{r}=\frac{\mathrm{2}{a}}{\:\sqrt{\mathrm{6}}}=\frac{\sqrt{\mathrm{2}}{a}}{\:\sqrt{\mathrm{3}}} \\ $$
Commented by mr W last updated on 17/Jan/24
$${what}'{s}\:{the}\:{edge}\:{length}\:\left({b}\right)\:{of}\:{the} \\ $$$${regular}\:{circumtetrahedron}\:{for} \\ $$$${the}\:\mathrm{4}\:{spheres}\:{with}\:{radius}\:{r}? \\ $$
Commented by mr W last updated on 17/Jan/24
Commented by mr W last updated on 17/Jan/24
$${i}\:{got}\:{b}=\mathrm{2}\left(\sqrt{\mathrm{6}}+\mathrm{1}\right){r} \\ $$