Magic
Triangles, or the Area Paradox

Cut
out a square of 12 units per side into 8 triangular pieces,
as shown below (fig.1). We can rearrange these 8 right triangles
together to form some square variants. Further below you
will find 4 possible configurations. The first configuration
is a real square (fig. 2.a); the second one (fig. 2.b), though
it seems to be a square is not a square at all! Can you say
why this is so? The last examples (fig. 2.c and 2.d) are
squares but with extra protruding triangular elements! Compare
them to the square of the fig. 2.a). What is wrong?