Mathematics applications with the Tangram puzzle

  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!

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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...

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  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

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The 13th shape...

  Here below, you'll find 12 convex polygons you can form with the Tangram puzzle. There is a 13th convex polygon, which is it? We mail you the solution if you send us a postcard from your country! (our address: Archimedes' Lab, Marie-Jo Waeber, CP1700, 16121 Genova Centro, Italy)