Geometric dissection theory A few days ago, i realized that one way to prove the Pythagorean Theorem is to dissect the given right-angled triangle into 2 triangles similar to it, and apply well-known properties of ratios of areas.
So this fundamental theorem can be viewed as a consequence of the possibility to dissect a given right-triangle into 2 similar sub-triangles. 
My question is: Which branch of geometry systematically studies dissections of general polygons (or other shapes) into other sub-polygons (sub-shapes) of a given type, and the metrical properties that emerge as consequences of these dissections? 
My search is on-going, i have seen some relevant results & books, for example the problem of dissecting a random triangle into n similar triangles, and books on discrete & combinatorial geometry that border on the subject (but not studied them yet), so i'm asking for some further guidance towards a systematic exposition of the relevant theory, in 2D, 3D or more dimensions..
 A: You might search for 'scissors congruence', which discusses which polyhedra (= higher-dimensional analogues of polygons) can be cut up to make which other ones.  The question of whether you can cut a cube into finitely many pieces and reassemble to make a regular tetrahedron of the same volume was on Hilbert's problem list of around 1900, and was one of the first to be solved (when Dehn showed that it can't).  I am not sure that this is exactly the sort of thing that you are looking for, but it is related and it's an interesting area of mathematics.  
A: The following paper proves (1) that every polygon may be partitioned (dissected)
into (a finite number of) non-obtuse triangles, and (2) into acute triangles:

Maehara, Hiroshi. "Acute triangulations of polygons." European Journal of Combinatorics 23, no. 1 (2002): 45-55.
  
            
  


Dissections specifically into similar triangles:

Jones, C. A., P. Jones, and A. B. Bolt. "Dissections of triangles into five similar triangles." The Mathematical Gazette 82, no. 494 (1998): 225-234.
  
            
  


See also:


*

*MathWorld on Triangle Dissection.

*The textbook Discrete and Computational Geometry has a chapter on polygon dissections.

*MO question: 3-piece dissection of square to equilateral triangle?

*MO question: Inside-out polygonal dissections. 

