Laying the Triangle Illusion "Mystery" to Rest.

Today, I was browsing through StumbleUpon, and I ran across this popular illusion that has been around for a while. The illusion is, that there are four shapes (as shown),


that when arranged differently form the same shape, but with a different Area.


Comparing the upper and lower triangles (we’ll call them X and Y, respectively), we can see that X has an empty square that seemingly disappears when rearranged into Figure Y, yet at a glance, the size of X and Y in their entirety seem to be identical. However, since the area of the individual shapes have not changed, this simply cannot be possible! So let’s break it down and see exactly what is happening in between arrangements.


Our goal here is to prove that the area of triangles X and Y are not equal. There are a few different ways that we can do this, but what we will do in this case is calculate areas A and B outside of the triangle. If these triangles are, in fact, identical, then areas A and B should be exactly equal.

We’ll calculate this with the standard formulas for area:

  • Rectangle: A = bh
  • Triangle: A = 1/2bh


We’re interested in the total area of the colored shapes above. We now have our totals for both Areas A and B. We can see that in triangle X, where there is the unexplained square, the outer area “A” is 32. In triangle Y, outer area “B” is 33, the difference of 1 square inch caused by the extra space in X.

Why this illusion is so effective at first glance, is that our eyes assume that the hypotenuse of X or Y as a whole is a straight line. In actuality, the slopes of the two aligning triangles are slightly different. In X, the line bends out slightly outwards, while in Y, the line retracts inward. Just enough to make that 1 square inch difference.

This dose of geekery brought to you by 2:30 AM, and the addictive power of StumbleUpon. If you found this informative, please Stumble it. :)


15 Responses to “Laying the Triangle Illusion "Mystery" to Rest.”

  1. 1 Jennifer Blake
    April 3, 2008 at 10:34 PM

    umm….yeah…..you need a girlfriend :-)

  2. 2 Anonymous
    May 22, 2008 at 3:02 AM

    “umm….yeah…..you need a girlfriend :-)”

    Just because someones smart and can figure things out? Who says he doesn’t have a girlfriend.

    It’s people like you who hate on others because your jealous of what they can do so you try and make them feel bad.

    I would way rather be excellent at math then probably have a slut like u/ur gf.

  3. 3 Ann Hiro
    May 22, 2008 at 3:59 AM

    Ms. Jennifer Blake, the gentleman in table 2 wants his sammich

  4. 4 Brad
    May 22, 2008 at 6:11 AM

    I agree with anonymous. Jennifer Blake you need to grow the **** up

  5. 5 Kevin
    May 22, 2008 at 9:54 AM

    Um, guys? She’s my sister, it’s alright, haha.

    Also curious, where’s everyone getting linked from? I’ve gotten around 150 hits on this page so far today.

  6. 6 Jennifer Blake
    May 22, 2008 at 10:00 AM

    lol its ok i’m just teasing him, thanks kevin

  7. 7 Kevin
    May 22, 2008 at 10:03 AM

    But the question now, Jennifer: Is your gf a slut? Just askin’.

  8. 8 Jennifer Blake
    May 22, 2008 at 10:05 AM


  9. June 5, 2008 at 10:35 AM

    Kevin, if you go to statcounter.com, you can get free stats that tell you where someone came from to find your page. That’s what I use for mine. :-)

  10. 10 Kevin
    June 5, 2008 at 9:18 PM

    Cool. I’ll check it out! :)

  11. July 3, 2008 at 10:19 AM

    Are you ever going to write a new post? :) lol

  12. 12 Kevin
    July 3, 2008 at 10:36 AM

    lol, one of these days…

  13. 13 john doe
    November 22, 2008 at 6:41 AM

    if you crop those shapes and calculate their areas separately, the areas won’t change if you arrange the shapes differently…you can you use cardboard or something…and then try the theory in practise

  14. 14 john doe
    November 22, 2008 at 7:24 AM

    here am I again…..i croped tho whole triangle, then I cropped tho hole and then arranged it to be the same triangle…peace….i went to kill myself.. :)

  15. 15 shipoopi
    October 15, 2012 at 10:30 PM

    lol this is some trippy stuff man hahahahahaha this is bs but its not grrr this is unbelievable

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