The length, breadth, and height of a cuboid are in the ratio 6 : 5 : 4,

Let the length, breadth, and height of the cuboid be in the ratio 6 : 5 : 4.

So take
Length = 6x
Breadth = 5x
Height = 4x

Volume of a cuboid:

$V = l \times b \times h$

$= (6x)(5x)(4x)$
$= 120x^3$

Given volume:

$120x^3 = 810000\sqrt{2}$

$x^3 = 6750\sqrt{2}$

$x = 15\sqrt{2}$

So the dimensions are:

Length = $6x = 90\sqrt{2}$ cm
Breadth = $5x = 75\sqrt{2}$ cm
Height = $4x = 60\sqrt{2}$ cm

Surface area of a cuboid:

$S = 2(lb + bh + hl)$

$= 2[(90\sqrt{2})(75\sqrt{2}) + (75\sqrt{2})(60\sqrt{2}) + (60\sqrt{2})(90\sqrt{2})]$

$= 2[(90 \times 75 \times 2) + (75 \times 60 \times 2) + (60 \times 90 \times 2)]$

$= 2[13500 + 9000 + 10800]$

$= 2(33300)$

$= 66600$

Surface area = $66600\ \text{cm}^2$