Equidecomposable Polygons
Based on web materials in Portuguese by Maria Alice Gravina

translated usingBabel Fish Translation  from Alta Vista

Simple and interesting theorems in Mathematics exist, through which one can develop a good part of  the contents that are part of the programs of our schools. If we have a clear-cut objective to be reached, in the case the demonstration of an interesting result, certainly the development of concepts and properties becomes much more significant, and with this the pupils learn with enthusiasm.

An example of this is the following theorem:

"If two polygons have the same area then it is always possible to decompose one of them into lesser polygons from which one can compose the other."

In other words, we can decompose the two polygons in lesser polygons, which are congruent to each other. This means that the two polygons can be decomposed equally, and by this they are called "equidecomposable polygons."

In some situations, depending on the form and the sizing of the polygons, we can discover an equidecomposition.For example easily, in the pairs of polygons below:

square and rectangle such that one of the sides of 
rectangle is the double of the side of the square 
parallelogram and rectangle with same bases and heights 

To see equidecompositions click here.

In other situations one such equidecompositions is not obvious. For example, in the situations below:

square and rectangle with same area 
square and hexagon with same area 

How to get a equidecomposition for these pairs of polygons?

To develop the theorem we go, step to the step, to construct a puzzle. The material presented here can be used in diverse ways, depending on the audience:

* A playful approach - they are puzzles that transform triangles into rectangles, rectangles in squares, two squares in an only one squared and a polygon into a square .
* A intuitive approach - triangles, squares, rectangles, parallelograms, parallelism, perpendicularity are some of the concepts developed in the construction of the puzzles. Here the geometric drawing with ruler and compass is important tool, since the precision of the figures is basic in the assembly of the puzzles.
* A deductive approach - the demonstrations are worked that  guarantee that the puzzles are mathematically correct. For this properties of angles and parallelism are used, of congruence and similarity of triangles, length and area.

This theorem was demonstrated by F.Bolyai in 1832 and, independently, in 1833 for G.Gerwien, an amateur German mathematician. F.Bolyai was the father of the mathematical celebrity Hungarian Janos Bolyai, creator of the Hyperbolic Geometry (also created by Lobatchevski and Gauss. )
It is natural to ask if the analogous result  is true for polyhedrons. Max Dehn, pupil of Hilbert, proved in 1900 that this is not true:  In particular, a regular tetahedron  and a cube of same volume are not equidecomposable.

The puzzles, step to the step: