Hey, what's going on with this triangle (geometry question)?

A friend of mine sent me a link to this picture. I looked at it for a little while, and can’t figure out what’s going on. Anyone have any insight?

For what it’s worth, I believe you can do something similar with a square. If f[sub]n[/sub] denotes the nth Fibonacci number, and you have an f[sub]n[/sub] by f[sub]n[/sub] square, you can cut it up to form an f[sub]n+1[/sub] by f[sub]n-1[/sub] rectangle. Presumably this is the same phenomenon–what is it?

Think slope. You know, y=mx+b.


But, judging by the graph paper, the slope is the same on both of the triangles.

Try this old GQ thread:

Tough math puzzler


Umm, now for the correct thread :slight_smile:

All of those Math Majors out there. Help me figure this out.


That’s an F for effort, NerdBoy. Don’t “judge by the graph paper,” calculate the slope.


This one is so old it has whiskers on it. I’ve seen it many times.

The area of the missing square is the same as the area of the black lines that delineate the various colored shapes. When the shapes are rearranged the right way, there appears to be a missing square, but it’s just a result of the rearrangement.

No, it isn’t even a straight edge. Count the lengths of the sides of the little triangles, and divide.

NO!! It has absolutely nothing to do with the size of the black lines.

The two small triangles have different slopes. The larger “triangles” formed when the shapes are put together are not triangles at all, they’re quadralaterals.


I see it now. My ability to think rationally was, once again, clouded by my vision.

Too expand on hajario (and, BTW, I got this wrong in one of the old threads) all you need to do is calculate the slope of the (apparently) similar triangles. You will see they are not similar.

Red traingle = 8 over by 3 up. Slope = 8/3 = 2.666…

Green triangle = 5 over, 2 up. Slope = 5/2 = 2.5.

Thus the “big” triangle is not a triangle at all. The two portions have very slightly different slopes. You are dealing with a four-sided figure.

Thanks, everyone.

This was discussed less than a month ago in the GQ thread Can anyone explain this Granted, the thread title isn’t very descriptive, but a search for triangle area turned it up.


There ya go…the answer, unveiled!!!

Just hold a straight edge against the screen and put it along the hypoteneuse of each of the shapes.

Hm. Well, that’s how I’ve seen it explained in a couple of places. I stand corrected.

Mea culpa. I must’ve screwed up my search somehow.

If you don’t have a straight edge or understand the equations, just look at the point where the two triangles meet. On the top figure, that point is on a grid point, but the same point on the lower figure is clearly within the red triangle. Likewise, the point where the two triangles meet on the bottom figure is also on a grid point, but the corresponding point on upper figure is clearly outside the figure.

Found a link you might enjoy, and interestingly enough, Fibonacci numbers don’t seem to be involved. Here is a link that turns a nxn square into a (n+1)x(n-1) rectangle.

This is in some ways similar to the Banach-Tarski paradox (which doesn’t “cheat” like these others do, with triangles that ain’t triangles, and squares that ain’t squares, and is much more impressive, too, IMO).

The Banach-Tarski theorem says, in effect, that you can take a ball the size of a BB, cut it up into finitely many pieces, then put those pieces back together to form a ball the size of the Sun!

The only catch is that those pieces are very bizarre. In fact, they’re so pathological that you can’t even define a volume for them.

Here’s a little webpage describing it briefly: