# Why does rearranging the pieces of this triangle illusion give a different area? [duplicate]

Possible Duplicate:
How come 32.5 = 31.5? ## marked as duplicate by Qiaochu YuanMar 7 '11 at 12:04

• fairly sure there was a duplicate of this question... – InterestedGuest Mar 2 '11 at 0:18
• There might be. That's what I thought and tried to find it (put very little effort) but couldn't. – 0x0 Mar 2 '11 at 0:20
• @InterestedQuest, @Sunil This was the dupe: math.stackexchange.com/questions/287/how-come-32-5-31-5 – Uticensis Mar 7 '11 at 11:24

If you count carefully, you'll see that the base is meant to be $13$ units long, while the height is $5$ units long. That means that the triangle on the top on the top figure, which has a height of $2$, should have a base of length $b$, where $$\frac{b}{2} = \frac{13}{5}$$ or $b = \frac{26}{5}$, longer than the $5$ units depicted.

Likewise, the bottom red triangle, with a base of size $8$ should have a height of length $h$, with $$\frac{h}{8} = \frac{5}{13}$$ or $h = \frac{40}{13}$, which is a little longer than the $3$ depicted.

So in fact, the "missing square" comes from misdrawing the pictures (or from having the individual figures drawn correctly, but the composed figures not being real triangles; the two inner triangles are not similar, though they "should" be).

Eye trick! Look at the angles formed where the red and green triangle meet.

• Looking at angles may not be enough given the thick lines. But the slope of the hypotenuse of the red triangle is $3/8 = 0.375$ while the slope for the green triangle is $2/5 = 0.4$. If the whole thing was a triangle (which it is not) then the slope would be $5/13 =0.3846\ldots$ – Henry Mar 1 '11 at 23:41
• Indeed the thick lines are what make the "angles different" - and which is what makes it sloppy. – milcak Mar 1 '11 at 23:47