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Mathematics applications with the Tangram puzzle

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Math relations between the 5 shapes
which form the Tangram puzzle

by G. A. Sarcone & M. J. Waeber


tangram sections

  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!

tangram formula
©1992-2007, 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:

Pb / Pa = 2(4a+b)/2(a+3b)

By substituting the value "b" with "asquare root2", we obtain: 1.0327...

tangram problem
©1992-2007, 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 Chuan-Chih 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. 596-599). 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!
12 tangram shapes
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