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Using 2 triangles each with base of 8 and height of 3, and 2 trapezoids with heights of 3 on top, 5 on bottom and height of 5, these four figures can create an area with 64 units squared. However, when rearranged as a rectangle with 13 x 5=65, one additional unit squared seemed to have been created. How is this possible?

image

Goodwin Lu
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Here's a slightly less subtle "demonstration" that a rectangle with area 5 can be rearranged into a rectangle with area 6. enter image description here enter image description here

grand_chat
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This is a classic illusion based on the Fibonacci number identity $$ 13 \times 5 = 1 + 8 \times 8 . $$

The "diagonal" of the rectangle isn't one. The slopes on each segment don't agree. There's one unit of area between the "diagonals".

Ethan Bolker
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unlike the wikipedia article, the hypothenuse is completely and utterly straight

No, it's not. Consider the bottom-left corner of the rectangle.

  • Let $\alpha$ be the angle in the yellow triangle, then $\tan \alpha = 3/8\,$.
  • Let $\beta$ be the angle in the green trapezoid, then $\tan \beta = 5/(5-3)=5/2\,$.

But then $\,\tan \alpha \tan \beta = 15 / 16 \ne 1\,$, so $\,\alpha+\beta \ne 90^\circ\,$ i.e. the two angles do not add up to a right angle. The slopes of the two hypotenuses differ by $\,90^\circ - \arctan 3/8 - \arctan 5/2 \simeq 1.25 ^\circ$.

dxiv
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  • In fact, not only is the "hypotenuse" not straight, it is bent in _exactly the same way_ as in the Wikipedia article: part of the "hypotenuse" has slope 2/5, and the rest of it has slope 3/8. – David K Sep 02 '17 at 00:57
  • @DavidK Indeed, and also thanks for the pointer to that cute animation in the duplicate link. Somewhat related: [the perpetual chocolate](https://math.stackexchange.com/a/743074). – dxiv Sep 02 '17 at 01:08