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

.