According to the question,

Length of the rectangular surface $=6m=600cm$

Breadth of the rectangular surface $=4m=400cm$

Height of the perceived rain $=1cm$

Then,

Volume of the rectangular surface $=l*b*h$

$=600*400*1c{{m}^{3}}$

$=240000c{{m}^{3}}$ …………….. (i)

It is also given that,

Radius of the cylindrical vessel $=20cm$

Assume the height of the cylindrical vessel be taken as h cm

We know that,

Formula for volume of the cylindrical vessel $=\pi {{r}^{2}}h$

$=\pi \times {{20}^{2}}\times h$ ……….. (ii)

As all the rain water is transferred to the cylindrical vessel

We can equate both (i) and (ii) for equal volumes,

$240000=\pi \times {{20}^{2}}\times h$

$h=190.9cm$

Therefore, the height of the cylindrical vessel nearly $191cm$.