How to prove this hypothesis on convex geometry viewed orthogonally down cartesian axes?

I don't claim to be a mathematician, and this could have admittedly been done to death (in which case, I'd appreciate a name/link!), but I'm trying to prove a concept for an imaging system I'm writing software for, and could do with some help mathematically proving the following theory:

Take an object of convex shape (not necessarily uniformly),
viewed from three directions to produce orthogonal (no perspective)
2D projections down the Cartesian (X, Y, Z) axes.

The sum of the areas produced by the object in each of the three
projections is constant, regardless of the orientation of the object.

So basically, in laymans terms; the object creates the same total profile in those three images regardless of how it is oriented.

The above hypothesis has been reached intuitively and could be wrong (in which case, proof of that would also be appreciated); but if it IS correct, how would I go about proving it mathematically?

I realise it could be easily proved for basic shapes (cubes etc) with a bit of trigonometry; but how could I prove it as a general case for convex objects of any shape?

I'm afraid, your theory is wrong. Imagine a $1\times 1\times 1$-cube with its edges parallel to the axes. For each projection, the area is $1$. This makes a sum of $3$. Now rotate the cube by $45^\circ$ along the $z$-axis. When projecting onto the $xy$-plane, you still get an area of $1$. For the other projections, you get $\sqrt{2}$ each. This makes a total of $1 + 2\sqrt{2} \neq 3$.