Comments on: Another Geometry Optical Illusion

By: willy

willy — Thu, 02 Feb 2012 17:15:16 +0000

ok. try this… much easier: inverting the colors you can see the line that conforms the triangle is not straight .. in white you can see the small curvature and if you draw a straight line dividing the figure number two, you´ll see the straight line doesnt fit straight. XD solve by paint

By: Andrew

Andrew — Sun, 27 Nov 2011 08:48:09 +0000

Why not just use formulas that we learned in 6th grade? Get the area of the triangles (a=1/2bh) and then split the quadrangles into a triangle and a rectangle (a=bh) The 2 quadrangles split into a rectangle with an area of 15, and triangles with an area of 5. Then you take the peach colored triangles, and do the math. a=1/2bh. a=1/2(3*8). a=12. now add everything up and you get 15+15+12+12+5+5=64. It works for both.

By: yupweak

yupweak — Wed, 21 Sep 2011 16:24:19 +0000

No, the point IS to trick the eye, and it’s NOT the same shapes in both of them. That is why it is weak. The premise isn’t even true.

By: Abisso

Abisso — Thu, 04 Aug 2011 10:59:59 +0000

Ok, dunno if it’s already been solved, but I’m pretty sure I’ve figured it out. Some basics first: congruent means “which has the same shape and size; area is a measure of the extent of a two-dimensional surface.
Let’s start from the left composition, which is made by 2 congruent (same shape and size) trapezoids and 2 congruent triangles. Using the squared sheet, we can see that each trapezoid has an area of 20 ((upper base + lower base) * height / 2) and each triangle has an area of 12 (base * height / 2). 20+20+12+12=64. This confirms what’s already written in the picture.

Now let’s take a look at the 2nd composition, the one on the right. We still see the 2 trapezoids, and what looks like to be the same 2 triangles, highlighted in yellow. But if you look closely, these are NOT the 2 triangles. The hypotenuse is not a segment at all, it’s “bent”, not straight. So, actually, in this composition, we can see: the 2 previous trapezoids; 2 small congruent triangles; 2 small congruent trapezoids.
Small triangle’s area is 5 ((2+5)/2).
Small trapezoid’s area is 7,5 ((2+3)*3/2).
5+5+7,5+7,5+20+20=65.

So, there really is no mistery at all: the two compositions use different shapes.

By: kathleen b.

kathleen b. — Wed, 03 Aug 2011 14:23:37 +0000

i don’t like this one cuz’ i’m BAD at math. i hate math anyway.

By: Melanie Blakeman

Melanie Blakeman — Fri, 29 Jul 2011 05:24:01 +0000

Yep that’s how I confirmed what I saw:)) It isn’t very complicated. The elaborate explanations truly do surprise me.

I like mathematics, but I’ve forgotten most of what I learned. I kept an old University primer textbook and look things up when it’s necessary, but here you see what is happening and the numbers confirm it:))

By: Elaine

Elaine — Tue, 26 Jul 2011 08:27:32 +0000

this one is really simple.they are of the same shape and areas.but they are used differently.as in.they are used and joined in different ways.therefore.it changes the overall surface.the overall surface will always be different if you put it together in different ways.

By: Jackson

Jackson — Sun, 26 Jun 2011 03:32:30 +0000

The point’s not to trick the eye, it’s the fact that they’re the same shapes in both of them, and the area’s different when arranged differently

By: Uncle Bob

Uncle Bob — Tue, 07 Jun 2011 23:41:38 +0000

Thank you yenyen for calculating the slopes, i would do it myself but I’m too lazy. You can see with a naked eye that in the first shape two yellow (or orange) triangles have a steady slope of 1/3, and in the second shape you can spot the change of course in their slopes. I also thank you for reinstating the perimeter to its rightful place in this problem. And can someone explain to me what the Hell is the “quadrangle”.
At the first look i could tell that it was a right-angle trapezoid, but nevertheless I have researched this “quadrangle” confusion. What I found was that quadrangle is used in architecture as a space or courtyard, usually rectangular (square or oblong) in plan, the sides of which are entirely or mainly occupied by parts of a large building. “Quadrilaterals” is often used in geometry, but it refers to a family of shapes that consist of four angle and sides. So those gray thingies are definitely are right-angle trapezoids!

By: Tom Bailey

Tom Bailey — Fri, 20 May 2011 16:28:52 +0000

The ppl that think this is just an illusion are incorrect. Kadri is also incorrect. Having two pieces of equal perimeter can result in them having different areas, but we are told that the pieces used are the SAME, and thus the areas MUST be the SAME. Since they aren’t, this is a sure bet something is wrong. Kudos to the ppl that used math to prove this. Not that math was needed to know that something was mischievious, but its good to know some ppl actually know how to do math.

By: yenyen

yenyen — Fri, 13 May 2011 05:49:27 +0000

check the slope along the diagonal in the rectangle.

it is not consistent. 1:3, 1:2, 1:3, 1:2, 1:3. meaning, it goes over one, up 3, then over one up 2, and so on.

a straight line would have a 1 over:3 up slope all the way.

lol at the circumference and ashhole comments. i hope you are kidding. also, i’m pretty sure circumference is reserved for circles and perimeter is the word you’re looking for (about this, i may be wrong; it’s been awhile since i’ve used math terms).

By: Melanie Blakeman

Melanie Blakeman — Sun, 20 Mar 2011 09:28:13 +0000

The so called hole, or thick line isn’t either. It’s two lines butted together. Kadra is correct. It’s obvious at first glance!!!

By: Mykiey

Mykiey — Sat, 12 Mar 2011 21:49:16 +0000

But that would still require the circumference to be slightly larger, you are both quazi right.

By: Andrew

Andrew — Wed, 01 Dec 2010 22:06:36 +0000

Kadri, you are incorrect. The answer is, as others have stated, that there is a small, narrow, long whole in the middle of the square.

This is also covered up by the extremely thick lines they used.

This is one of a class of puzzles where lines that are ALMOST parallel are matched up and the human mind assumes they are.

It is very easy to check by cutting up some white graph paper and re-arranging it on a blue tablecloth

By: Kadri

Kadri — Sat, 20 Nov 2010 23:32:32 +0000

Actually a fact that many people don’t seem to know is that any rectangles area is related to its circumference (the most efficient rectangle being the square). The bigger the circumference the bigger the area but they don’t increase equally. So it is possible that the areas of the small parts that make the square remain the same but if you place them to make a rectangle (the circumference increases) then the total area is bigger by one unit. It isn’t an optical illusion, it is basic maths.