Triangle Dissection Paradox


The above two figures are rearrangements of each other, with the corresponding triangles and polyominoes having the same areas. Nevertheless, the bottom figure has an area one unit larger than the top figure (as indicated by the grid square containing the dot).

The source of this apparent paradox is that the "hypotenuse" of the overall "triangle" is not a straight line, but consists of two broken segments. As a result, the "hypotenuse" of the top figure is slightly bent in, whereas the "hypotenuse" of the bottom figure is slightly bent out. The difference in the areas of these figures is then exactly the "extra" one unit. Explicitly, the area of triangular "hole" (0, 0), (8, 3), (13, 5) in the top figure is 1/2, as is the area of triangular "excess" (0, 0), (5, 2), (13, 5) in the bottom figure, for a total of one unit difference.

See also

Curry Triangle, Dissection Fallacy, Tangram Paradox

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Bogomolny, "A. Fibonacci Bamboozlement.", R. "Harder Fibonacci Puzzles."

Referenced on Wolfram|Alpha

Triangle Dissection Paradox

Cite this as:

Weisstein, Eric W. "Triangle Dissection Paradox." From MathWorld--A Wolfram Web Resource.

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