Vanishing Square

The diagrams below show two arrangements of the same four pieces. Both form a 13 by 5 right triangle. Yet the first arrangement contains a 1-square-unit gap, while the second does not. Where did the additional area go?

Here’s the first arrangement:

And here’s the rearranged version without the gap:

Solution

Neither arrangement actually forms a true triangle. The “hypotenuse” isn’t a straight line in either case—it’s slightly bent, but the bend is so subtle that it’s hard to notice without close inspection.

The key is in the slopes of the two triangular pieces:

Big triangle: 3 units tall by 8 units wide, giving a slope of $\frac{3}{8} = 0.375$ . Small triangle: 2 units tall by 5 units wide, giving a slope of $\frac{2}{5} = 0.400$ .

Since $\frac{3}{8} \neq \frac{2}{5}$ , the two triangular pieces don’t have the same slope. When placed together, they create a very slight bend in what appears to be the hypotenuse.

In the first arrangement, the “hypotenuse” bends very slightly inward, while in the second arrangement, it bends very slightly outward. This gap accounts for the 1 unit² of area difference.

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