Area of a Surface Formed by Mutually Perpendicular Surfaces

The diagram below shows a small cuboid with one corner at the origin, having sides of length
$dx,dy,dz$
.

The area of the side in the
$xy$
plane is
$S_1 =dxdy$

The area of the side in the
$yz$
plane is
$S_2 =dydz$

The area of the rectangle formed by the diagonal shown is
$S =dy \sqrt{(dx)^2 +(dz)^2}= \sqrt{(dxdy)^2 +(dydz)^2}$

Then
$(dA_)^2 = (dS_1)^2 + (dS_2)^2$

We can extend this to three dimensions by noticing the surface of the cuboid in the
$xz$
plane is perpendicular to the diagonal surface above. Using the reasoning above for perpendicular areas gives
$(dA)^2 =(dS)^2 + (S_3)^2 = (dS_1)^2 + (dS_2) + (dS_3)^2$
.