# How can this "illegal geometry" problem be possible? [duplicate]

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? • This is the missing square puzzle. Sep 2, 2017 at 0:04
• so the second figure has triangles slightly bigger than the triangles in figure one? Sep 2, 2017 at 0:06
• also, unlike the wikipedia article, the hypothenuse is completely and utterly straight: prntscr.com/gg199b Sep 2, 2017 at 0:10
• @GoodwinLu then it doesn't work, the truth is it's not possible it's an illusion.
– user451844
Sep 2, 2017 at 0:14
• According to the figure on the right, $\frac25=\frac38$. Who knew? Sep 2, 2017 at 1:41

Here's a slightly less subtle "demonstration" that a rectangle with area 5 can be rearranged into a rectangle with area 6.  • Nicely done (+1).
– dxiv
Sep 2, 2017 at 0:47

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".

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 $\,90^\circ - \arctan 3/8 - \arctan 5/2 \simeq 1.25 ^\circ$.

• 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. Sep 2, 2017 at 0:57
• @DavidK Indeed, and also thanks for the pointer to that cute animation in the duplicate link. Somewhat related: the perpetual chocolate.
– dxiv
Sep 2, 2017 at 1:08