# Hey, what's going on with this triangle (geometry question)? **URL:** \ **Category:** Factual Questions\ **Created:** [September 24, 2001, 2:41am UTC](https://boards.straightdope.com/t/hey-whats-going-on-with-this-triangle-geometry-question/83444 "2001-09-24T02:41:40Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [September 24, 2001, 2:41am UTC](https://boards.straightdope.com/t/hey-whats-going-on-with-this-triangle-geometry-question/83444/1 "2001-09-24T02:41:40Z")
A friend of mine sent me a link to [this picture](http://www.simeonmagic.com/triangle/triangle1.htm). 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? ---
**Author:** ![hajario](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hajario/32/171_2.png) [@hajario](https://boards.straightdope.com/u/hajario)\ **Post date:** [September 24, 2001, 2:42am UTC](https://boards.straightdope.com/t/hey-whats-going-on-with-this-triangle-geometry-question/83444/2 "2001-09-24T02:42:45Z")
Think slope. You know, y=mx+b. Haj ---
**Author:** ![NerdBoy](https://avatars.discourse-cdn.com/v4/letter/n/d26b3c/32.png) [@NerdBoy](https://boards.straightdope.com/u/NerdBoy)\ **Post date:** [September 24, 2001, 2:45am UTC](https://boards.straightdope.com/t/hey-whats-going-on-with-this-triangle-geometry-question/83444/3 "2001-09-24T02:45:15Z")
But, judging by the graph paper, the slope is the same on both of the triangles. ---
**Author:** ![Arjuna34](https://avatars.discourse-cdn.com/v4/letter/a/eb8c5e/32.png) [@Arjuna34](https://boards.straightdope.com/u/Arjuna34)\ **Post date:** [September 24, 2001, 2:47am UTC](https://boards.straightdope.com/t/hey-whats-going-on-with-this-triangle-geometry-question/83444/4 "2001-09-24T02:47:40Z")
Try this old GQ thread: [Tough math puzzler](http://boards.straightdope.com/sdmb/showthread.php?threadid=33894) Arjuna34 ---
**Author:** ![Arjuna34](https://avatars.discourse-cdn.com/v4/letter/a/eb8c5e/32.png) [@Arjuna34](https://boards.straightdope.com/u/Arjuna34)\ **Post date:** [September 24, 2001, 2:49am UTC](https://boards.straightdope.com/t/hey-whats-going-on-with-this-triangle-geometry-question/83444/5 "2001-09-24T02:49:03Z")
Umm, now for the correct thread 🙂 [All of those Math Majors out there. Help me figure this out.](http://boards.straightdope.com/sdmb/showthread.php?threadid=40963) Arjuna34 ---
**Author:** ![hajario](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hajario/32/171_2.png) [@hajario](https://boards.straightdope.com/u/hajario)\ **Post date:** [September 24, 2001, 2:49am UTC](https://boards.straightdope.com/t/hey-whats-going-on-with-this-triangle-geometry-question/83444/6 "2001-09-24T02:49:11Z")
> [@](#): > > \*Originally posted by NerdBoy \* > \*\*But, judging by the graph paper, the slope is the same on both of the triangles. \*\* That’s an F for effort, **NerdBoy**. Don’t “judge by the graph paper,” calculate the slope. Haj ---
**Author:** ![Scarlett67](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/scarlett67/32/7387_2.png) [@Scarlett67](https://boards.straightdope.com/u/Scarlett67)\ **Post date:** [September 24, 2001, 2:49am UTC](https://boards.straightdope.com/t/hey-whats-going-on-with-this-triangle-geometry-question/83444/7 "2001-09-24T02:49:18Z")
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. ---
**Author:** ![rjk](https://avatars.discourse-cdn.com/v4/letter/r/ed655f/32.png) [@rjk](https://boards.straightdope.com/u/rjk)\ **Post date:** [September 24, 2001, 2:51am UTC](https://boards.straightdope.com/t/hey-whats-going-on-with-this-triangle-geometry-question/83444/8 "2001-09-24T02:51:39Z")
No, it isn’t even a straight edge. Count the lengths of the sides of the little triangles, and divide. ---
**Author:** ![hajario](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hajario/32/171_2.png) [@hajario](https://boards.straightdope.com/u/hajario)\ **Post date:** [September 24, 2001, 2:55am UTC](https://boards.straightdope.com/t/hey-whats-going-on-with-this-triangle-geometry-question/83444/9 "2001-09-24T02:55:57Z")
> [@](#): > > \*Originally posted by Scarlett67 \* > \*\*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 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. Haj ---
**Author:** ![NerdBoy](https://avatars.discourse-cdn.com/v4/letter/n/d26b3c/32.png) [@NerdBoy](https://boards.straightdope.com/u/NerdBoy)\ **Post date:** [September 24, 2001, 3:01am UTC](https://boards.straightdope.com/t/hey-whats-going-on-with-this-triangle-geometry-question/83444/10 "2001-09-24T03:01:06Z")
I see it now. My ability to think rationally was, once again, clouded by my vision. ---
**Author:** ![friedo](https://avatars.discourse-cdn.com/v4/letter/f/8edcca/32.png) [@friedo](https://boards.straightdope.com/u/friedo)\ **Post date:** [September 24, 2001, 3:06am UTC](https://boards.straightdope.com/t/hey-whats-going-on-with-this-triangle-geometry-question/83444/11 "2001-09-24T03:06:07Z")
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. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [September 24, 2001, 3:25am UTC](https://boards.straightdope.com/t/hey-whats-going-on-with-this-triangle-geometry-question/83444/12 "2001-09-24T03:25:06Z")
Thanks, everyone. ---
**Author:** ![bibliophage](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bibliophage/32/7615_2.png) [@bibliophage](https://boards.straightdope.com/u/bibliophage)\ **Post date:** [September 24, 2001, 5:51am UTC](https://boards.straightdope.com/t/hey-whats-going-on-with-this-triangle-geometry-question/83444/13 "2001-09-24T05:51:07Z")
This was discussed less than a month ago in the GQ thread [Can anyone explain this](http://boards.straightdope.com/sdmb/showthread.php?threadid=84410) Granted, the thread title isn’t very descriptive, but a search for **triangle area** turned it up. ---
**Author:** ![Evnglion](https://avatars.discourse-cdn.com/v4/letter/e/f04885/32.png) [@Evnglion](https://boards.straightdope.com/u/Evnglion)\ **Post date:** [September 24, 2001, 11:38am UTC](https://boards.straightdope.com/t/hey-whats-going-on-with-this-triangle-geometry-question/83444/14 "2001-09-24T11:38:34Z")
> [@](#): > > \*Originally posted by NerdBoy \* > \*\*But, judging by the graph paper, the slope is the same on both of the triangles. \*\* hmmm…no > [@](#): > > \*Originally posted by friedo \* > \*[Snip]\*\*\*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.\*\* There ya go…the answer, unveiled!!! ---
**Author:** ![Mangetout](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mangetout/32/19_2.png) [@Mangetout](https://boards.straightdope.com/u/Mangetout)\ **Post date:** [September 24, 2001, 12:02pm UTC](https://boards.straightdope.com/t/hey-whats-going-on-with-this-triangle-geometry-question/83444/15 "2001-09-24T12:02:23Z")
Just hold a straight edge against the screen and put it along the hypoteneuse of each of the shapes. ---
**Author:** ![Scarlett67](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/scarlett67/32/7387_2.png) [@Scarlett67](https://boards.straightdope.com/u/Scarlett67)\ **Post date:** [September 24, 2001, 12:28pm UTC](https://boards.straightdope.com/t/hey-whats-going-on-with-this-triangle-geometry-question/83444/16 "2001-09-24T12:28:54Z")
> [@](#): > > \*Originally posted by hajario \* > **NO!! It has absolutely nothing to do with the size of the black lines.** Hm. Well, that’s how I’ve seen it explained in a couple of places. I stand corrected. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [September 24, 2001, 4:24pm UTC](https://boards.straightdope.com/t/hey-whats-going-on-with-this-triangle-geometry-question/83444/17 "2001-09-24T16:24:58Z")
> [@](#): > > \*Originally posted by bibliophage \* > \*\*This was discussed less than a month ago in the GQ thread [Can anyone explain this](http://boards.straightdope.com/sdmb/showthread.php?threadid=84410) Granted, the thread title isn’t very descriptive, but a search for **triangle area** turned it up. \*\* Mea culpa. I must’ve screwed up my search somehow. ---
**Author:** ![ZenBeam](https://avatars.discourse-cdn.com/v4/letter/z/3ab097/32.png) [@ZenBeam](https://boards.straightdope.com/u/ZenBeam)\ **Post date:** [September 24, 2001, 6:05pm UTC](https://boards.straightdope.com/t/hey-whats-going-on-with-this-triangle-geometry-question/83444/18 "2001-09-24T18:05:20Z")
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. ---
**Author:** ![23skidoo](https://avatars.discourse-cdn.com/v4/letter/2/f04885/32.png) [@23skidoo](https://boards.straightdope.com/u/23skidoo)\ **Post date:** [September 24, 2001, 7:00pm UTC](https://boards.straightdope.com/t/hey-whats-going-on-with-this-triangle-geometry-question/83444/19 "2001-09-24T19:00:08Z")
> [@](#): > > \*Originally posted by ultrafilter \* > \*\*A friend of mine sent me a link to [this picture](http://www.simeonmagic.com/triangle/triangle1.htm). 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? \*\* Found a link you might enjoy, and interestingly enough, Fibonacci numbers don’t seem to be involved. [Here](http://www.cut-the-knot.com/pythagoras/tricky.html) is a link that turns a nxn square into a (n+1)x(n-1) rectangle. ---
**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\ **Post date:** [September 25, 2001, 10:16am UTC](https://boards.straightdope.com/t/hey-whats-going-on-with-this-triangle-geometry-question/83444/20 "2001-09-25T10:16:27Z")
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: [http://www.math.hmc.edu/funfacts/ffiles/30001.1-3-8.shtml](http://www.math.hmc.edu/funfacts/ffiles/30001.1-3-8.shtml) [Next page](https://boards.straightdope.com/t/hey-whats-going-on-with-this-triangle-geometry-question/83444.md?page=2)