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  
\[(dA)^2 =(dS)^2 + (S_3)^2 = (dS_1)^2 + (dS_2) + (dS_3)^2\]
.

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