Heron's formula states that if you have a triangle $T \subset \Bbb R^2$of sides $a,b,c$ then the hypervolume of a right-angled hyper-parallelepiped (is there a better word for this) of sides $a+b+c,a+b-c,a+c-b,b+c-a$ is $4$ times the hypervolume of $T \times T$.
Is there a procedure to cut them up into a finite number of (polyedral, Banach–Tarski is not quite welcome here) pieces and rearrange them and/or add a finite number of identical pieces to both shapes, to show that they have equal hypervolume ?
For comparison I'm thinking of a geometrical proof of Pythagoras' indentity with a square inscribed in a another square.
I would want a procedure that's mostly independant on $a,b,c$ (requiring the triangle to be acute would be okay I guess), for example with endpoints whose coordinates are affine functions on $a,b,c$ and the coordinates of the triangles.