Brilliant Progressions

Let the edges be $\frac{a}{r},a,ar$ , where $r>1$

By question , $\frac{a}{r},a,ar=216$

$a^{3}=216$

$a=6$

Also , $2(\frac{a}{r}.a+a.ar+\frac{a}{r}.ar)=252$

$\frac{1}{r}+r+1=\frac{7}{2}$

$r=\frac{1}{2},2$

To find the longest side , $a=6$ , $r=2$

The length of the longest side is $ar=6\times2=12$

Hence , the answer is $12$ .