How can this "illegal geometry" problem be possible? 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?



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