I'm sure many would have came across the following puzzle.

How do we account for the missing unit? In both cases, it appears that the triangles are 13 units wide by 5 units tall. However, it's all but just an optical illusion! Let's solve it using a mathematical approach.
The four shapes give a total of 32 units of area (yellow = 7; red = 0.5 * 8 * 3 = 12; blue = 0.5 * 5 * 2 = 5; green = 8). However, the blue triangle has a width-height ratio of 5:2, while the red triangle has a wide-height ratio of 8:3. With different width-height ratios, how can their hypotenuses be of the same gradient? Thus, the apparent combined hypotenuse in each figure is actually bent!
If we take the area the large triangle formed with the four shapes, we get 0.5 * 13 * 5 = 32.5 units. So in the figure where the 4 shapes totalling 32 units seem to fill up the triangle completely, we can't account for the additional 0.5 units. Similarly, in the figure with a missing unit, the 4 shapes take up 32 units and there's a missing unit, giving rise to a total of 33 units. Again, we can't account for the missing 0.5 units. Adding the differences together, this explains for the 1 missing unit we see. On a sidenote, if we take the length of the hypotenuses of the 2 smaller triangles and sum them up, we get √(82 + 32) + √(52 + 22) = 8.544004 + 5.385165 = 13.929169. The hypotenuse length of the big triangle is √(52 + 132) = 13.928388. So the 2 smaller triangles gives a slightly longer hypotenuse of 0.000781 units, too small to be detected by the naked eye but significant enough to produce an optical illusion!
Intuitively, the top angle made by the hypotenuse with the side of 8 units, and the hypotenuse with the side of 13 units should be exactly the same angle. By the law of similar triangles, all the angles of all three triangles should be exactly the same. But let's take one that's immediately obvious:
sin
-1(5/13) = 22.619865 degrees
sin
-1(3/8) = 22.024313 degrees
Again, we see a mismatch of 0.595552 degrees between the component triangles and the larger triangle they are supposed to build. Just over half a degree is very hard to see by visual inspection and the presence of a thick border line covers the mismatch up nicely!
Labels: Interesting matters