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. :)

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umm….yeah…..you need a girlfriend :-)

“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.

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

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

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.

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

But the question now, Jennifer:

Isyour gf a slut? Just askin’.lol!!!

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. :-)

Cool. I’ll check it out! :)

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

lol, one of these days…

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

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.. :)

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