For sand-casting a steel rectangular plate with dimensions $80 \: mm \times 120 \: mm \times 20 \: mm$, a cylindrical riser has to be designed. The height of the riser is equal to its diameter. The total solidification time for the casting in $2$ minutes. In Chvorinov's law for the estimation of the total solidification time, exponent is to be taken as $2$. For a solidification time for $3$ minutes in the riser, the diameter (in mm) of the riser is ________ (correct to two decimal places)