Reading up on Cauchy's stress theorem, I have stumbled over the so-called Cauchy tetrahedron, which is an important part of the theorem's proof. The following is cited straight from Wikipedia, but a similar argument (albeit shorter) is given in Aris' Vectors, tensors, and the basic equations of fluid mechanics p. 101:
To prove this expression, consider a tetrahedron with three faces oriented in the coordinate planes, and with an infinitesimal area $\mathrm{d}A$ oriented in an arbitrary direction specified by a normal unit vector $n$. [...] The area of the faces of the tetrahedron perpendicular to the axes can be found by projecting $\mathrm{d}A$ into each face (using the dot product): $$ \mathrm{d}A_i = (n\cdot e_i) \mathrm{d}A $$
Can somebody give an explanation or better argument for this? I fail to see how this seems to be (trivially) true.
Here is a figure illustrating this (also taken from Wikikpedia):
Consider a two-dimensional shape of area $A$ with normal $n$. And imagine that you are looking directly at it (along $n$). You will observe it to have its true area $A$.
Now imagine you rotate the shape by an angle $\theta$ about some arbitrary axis so that it now points in the direction $n'$. If you are still looking along $n$ then you will observe the shape to have an area $A\cos\theta=A n\cdot n'$.
If you agree with that then the result in question should become clear.