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Math
relations between the 5 shapes
which
form
the
Tangram
puzzle
by G. A. Sarcone & M. J. Waeber


The
math relations between the 5 basic shapes of the Tangram puzzle
shown below will allow you to create and/or solve new interesting
geometric puzzles. The small isosceles triangle B, the
square C and the rhomboid D of the Tangram all
have the same area, but their surfaces are different.
Actually, the square C has the smallest perimeter (4a < 2(a+b)).
The large isosceles triangle E is 4 times the size of
the small isosceles triangle A, but curiously its perimeter
is only 2 times as big!

©19922007, Sarcone & Waeber, Genoa 
A
small difference...
The
geometrical shapes in fig. a) and b) seem to be identical...
It's obvious that they are made with the same 4 Tangram pieces,
but thanks to the table above, we can calculate the perimeter
of each geometric figure and find that the perimeter of second
diagram is approx. 1.03 times larger than the first one.
The math expression is:
P_{b} / P_{a} =
2(4a+b)/2(a+3b)
By
substituting the value "b" with "a2",
we obtain: 1.0327...
©19922007,
Sarcone & Waeber, Genoa

Can
you form a perfect trapezium as shown in fig. c below using
all 7 tan pieces? The table above will be helpfull to determine
wether the puzzle is possible or not...
Source: Almanacco
del Matematico, G. Sarcone, 2001

The
13th convex shape...
In
1942, Fu Traing Wang and ChuanChih
Hsiung (also known as Xióng Quánzhì,
in Chinese: 熊全治) proved that there are
exactly 13
convex tangram configurations (cf. "A theorem
of the tangram", The American Mathematical Monthly,
n. 49, pp. 596599). Well... Here below, you'll find 12 convex
polygons you can form with the Tangram puzzle. Can you guess
which is the 13th convex tangram configuration? Mail us your
suggestion!



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