Vanishing area paradox

Vanishing area paradox      

Geetha Ravindran

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?

fig.1

The 8 triangles of the Circea's Puzzle

 

The 4 possible configurations 

When we place side by side two right triangles of Circea's puzzle, a small one to a large one, we obtain another larger triangle which seems to be a right triangle. But in reality, according to the measures of the illustration 1.a), the angle a can not be equal to 90 degrees. In fact,
a = arctan 7/6 + arctan 5/6 = approx. 89.2 degrees

So, the inscribed rectangle of the squares 2.a) and 2.b) is actually a parallelogram (see fig. 1.b), and according to the way this one is oriented, it changes the square into an irregular octagon (fig. 3)! The fig. 4) is a visual way to demonstrate that the polygon in the fig. 2.b) can not be a square. We understand it at a glance! 

 

 

 

Reassembling triangular puzzle pieces, induces always to paradoxical conclusions. The squares of the fig. 2.c) and 2.d) have extra triangular elements. Is then their area larger than the one in the fig. 2.a)? As before, we have to consider the angles of each right triangle which form these squares. By doing that, we will easily notice that the hypotenuse slopes of the small and of the large right triangles are slightly different (a difference of approx. 0.8 degrees, visually unnoticeable). So, the 8 right triangles do not form exactly a square and the sum of all these tiny fitting errors (grey zones in the fig. 5) is equal to the area of the protruding triangular elements. In short, space apparition is only illusion!

 

Paradoxical missing square puzzle
(called 'Fehlendes-Quadrat-Puzzle', in German; and 'wigparadox', in Dutch)
We can even enhance the 'Vanishing area paradox' effect by adding 4 squares of 6 units per side to the Circea's puzzle. When you rearrange the puzzle pieces a square space appears or disappears (see examples below).

 

Both squares are formed with exactly the same 12 pieces.
The second one, however, needs an extra piece!