211 Replies to “Impossible Triangle Illusion no.2”

  1. I dont understand what answer should come. I’ve looked and looked also, and cant see what answer i should be finding.

  2. I did that in photoshop. and overlayed them. all that seems to do for me is confirm that all the pieces are exactly the same. i dont see what “the answer” is. and i dont understand how the area of a triangle can be reduced while all its lengths stay equal. SO CONFUSING!!!!!! :)

  3. *** SPOILER ALERT ***

    It’s easy…

    The question is (if you don’t understand the question…) Why is there an empty space, if the four small pieces are all the exact same size (when comparing the top to the bottom).

    The answer is that the two diagrams don’t take up the same amount of space elsewhere. It LOOKS like the top 4 pieces make up a triangle, when in fact it’s really a 4-sided figure. The diagonal “line” runnning along the top left (of the first diagram) is actually two line segments with a small inward bend. Two straight lines, but meeting at a wide angle (not quit 180 degrees, which would be a straight line.. more like 179 degrees (estimated))

    In the second diagram, it’s a small “outward bend” along that same edge. (it’s two straight lines again, but they do not run at the same slant relative to the page, so they meet at an angle.)

    The difference between that inward bend and the outward bend, when spread across that line, actually amounts to one whole square. (the math is easy on this, trust me)

    To further prove it, look at the point of inflection on the top diagram (where the red triangle touches the green triangle.) In the top diagram, it’s exactly on the interersection of lines on the grid. Now look at that same (X,Y) point on the lower diagram, which is now within the Red triangle. See how far it’s within the Red?

    In looking at the top diagram, your eye tends to assume that it’s one giant triangle made up of 4 pieces. But hold a piece of paper up, with the edge along that diagonal line, and you’ll see that it’s really not one straight line after all!

  4. Thank you! Stupidly enough i saw that bend in photoshop. But just thought it was a tiny mistake. Not enough to explain anything. How wrong i was :).
    Now i can sleep again

  5. There is actualy an analitical way to find out what’s wrong.

    The area of the “triangles” should be the sum of the areas of the objects composing them.

    Now, this sum is:

    2*5/2 = 5, for the green triangle
    3*8/2 = 12, for the red
    3*5 = 15 for the two L shaped objects.

    TOTAL AREA = 32

    The area of the big suposed triangle, however, is 5*13/2 = 32,5

    The only explanation is that the big object it is not a triangle.

    Other way would be to calculate the sin of the sharp angles for the green and red triangles. It’s not the same and this tells us there are two segments and not one straight line.

  6. Oh poop i alerjik 2 geomutry .But it still makes perfect sense and i would have fuigured it out if i wasnt so lazy

  7. I am ashamed to admit that it took me as long as 2 minutes to figure this out. The answer is so obvious and it will only take one VERY short paragraph to explain it. Comment # 10 is correct.

  8. o I got this one right away, its the transfunction of the x axis and the velocity of the y axis when put in a parallel conversion equilevent to the triangles centrifical force.

    Im just kidding ya, i was just trying to act smart

  9. Graphs with colored shapes = How it was made-Photoshop. PP is probably responsible for all the impossible objects.

  10. wow this only took me about 40 seconds to figure out! since the red and cyan triangles are different sizes the red one will ofcourse have to stretch out the two ther peices…..
    did that make sense???

  11. wow this only took me about 40 seconds to figure out! since the red and cyan triangles are different sizes the red one will ofcourse have to stretch out the two ther peices…..
    did that make sense??? oo if u like adventurequest then go here for a game that youll love

  12. Ok for all the people who STILL don’t get it, here’s a simple explenation:

    The longest side of the triangle is called the hypotenuse, but lets call it h. Ok, if h ISN’T straight, then it must be in a curve, right?
    Right. So, if it’s curved, then it’s either adding more space if it’s curved outward, or removing space if its curved inward.
    In the first diagram, h is curved inward, so there is some triangle missing. In other words, its smaller than it looks.
    In the second diagram, h is curved OUTWARD, so there is MORE triangle than it seems. In fact, there is exactly 1 sq. unit’s worth of square added.
    So, it’s really simple. If you STILL confused, email me at [email protected], and i’ll try to get some simple geometry into you!

  13. wow me and my friends have never figured this out for like a year….. pretty pathetic ( but i didnt have photoshop euther) now im gonna tell my homies

  14. The blue triangle on top makes the rectangle 3X5 (15 squares)
    The red triangle on top makes the rectangle 2X8 (16 squares)
    keep it simple stupid

  15. what’s the problem?
    simple math’…
    the light green + the red = 13 squers

    the light green + the orang + the darker green = 12=]

    it’s really easy…….

  16. its not so simple like everyone says it is but i will try to explain it.

    going back to the line not being straight, each seperate figure is made up of straight lines, however, the slopes of the two triangles are different. so lets just throw in some random number like the angle of the red one is 40 degrees and the angle of the green is 45 degrees. they now in the first one form a line going inward of the triangle. now take them and put them like they are in the second “triange” they form an outward line. this gives it the extra space to make that extra square.
    this is really explaining number 5 in a more sensible way.
    also try showing it to a math teacher, they might be able to explain it better.

  17. it has to do with the slopes of the different triangles (red and green). then after figuring out the slopes its along the lines of #5

  18. don’t listen to these idiot jokers who think they are being cute by giving long explanations about how it’s the curve of the lines

    it has nothing to do with the curve of the lines, you can take the top image into photoshop and rearrange the pieces and you’ll still come up with a space

  19. lol guys its like this
    convex / (larger)

    concave _/ (smaller)

    the triangles have different slopes, if you switch their places it’ll change the entire triangle from concave to convex, or vise versa

  20. I have constructed the shapes in Freehand, a with exact numeric values, no curve lines, and I still get the effect.
    I really don’t think that the answer is the one of the curved lines

  21. alright all this dumb talk about curved lines and such is plain stupid and too boring for me to read so if its right then whatever but i suspect its not so let me put in my speculations.

    From what i see obviously the answer is :the red triangle has been moved on top and the green one on bottom right? right. problem is the green one wouldnt be able to stretch out far enough to cover the same area as the red one does since its smaller but the answer to that problem is the orange non-triangle (dont know what you call it) which when placed where it is on the 2nd picture stretches out the green triangle far out enough to cover the same area. but in stretching itself out it leaves a little friggin square that started this dumb debate with even dumber people. it took me 5 minutes to figure this out, 3 of which were spent reading the other comments

  22. Ruben’s answer link is correct. If all of you morons who still think it has nothing to do with curved lines need prrof, check it out. You cannot make a perfect triangle and cut out the pieces and rearrange into another identical size triangle with a gap. Those of you who say you used paper, freehand, your mother’s pantyhose, are wrong. The triangles you are making are not identical.

  23. ok, I drew out the figures on graph paper using the exact square by square measurements. I cut them out and put them side by side yet they still seem to occupy the same amount of space. Draw the geometric figures on a sheet of graph paper useing a ruler and the exact measurements in the illusion. It still works even with a consistantly straight line. WTF!?!?!?

  24. Ok, this is the final answer whether the lines are straight or curves:

    if they where straight, the proportion would have to be:

    2:5 = 3:8
    dark green red
    =>2 = (3:8)*5
    =>2 = 15:8

    BUT: 15:8 is not 2…
    the haven’t the same proportion.
    => the left angle of the dark green triangle is bigger than the left angle of the red one.
    =>the first BIG triangle is concave, the second is convex.


  25. Ok, the so called authorities posted here have not yet impressed me with a plausible explaination.
    Is the illusion that the diagonal line isnt straight when it looks straight?
    I dont think so. and the difference makes another squaare? no , i think that is too easy. that would not follow. if there were more area in one than the other than it would be just that, one has more area because of the curved line.
    but i think it is more interesting than that.
    It appears that the two constructed trangles have the same amount of units x and y, assuming they are true triangles and that would mean they have the same area, However within the same area, the same peices reconfigured fit the same area as the larger constructed triangle. so the illusion is that i have a grasp of its reality.
    I printed it out
    cut out the top peices
    and they act just as you would expect
    they fit over the bottom and make a space
    but it the sum of the area of all the peices = the area of the larger construct, then where dows that extra space get suptracted from the sum of the little peices , just by being configured within the same area?

    even if there was no curve in the top diagonals
    i still think it would be the same result
    as yet i am not impressed by any explaination or fancy sources of official looking diagrams.
    i am still shaking my head over this.
    i think essentially , that even if the so called curved lines were truely straight, the effect would still be the same , so , that would blow the curved line theory out of the water.
    so the curved line dosnt explain where the blank square came from.
    Unless im really missing something….

  26. Some how it seems that the sum of the area of all the little peices does not necessarily = the total area of the larger constructed triangle shape, Why?!?
    I dont know!
    yeah the little square poped up and the same peices fit in the same area of the larger triangle construct.
    but if all the lines were true straight lines
    it still wouldnt explain the space,
    the real explaination i think is far mor brainier profesorial than any of us so far…

  27. Wow, all this pointless math equations and various theories about how they aren’t true triangles. You’re all looking at it too deeply, take a step back, will you?

    It took me a total of a minute to figure out what was going on here.

    It doesn’t lose it’s spacial area in the second picture because:

    1.)The red triangle’s height more than makes up for the loss of the orange non-triangle shape.

    2.)Although the horizontal length of the green triangle isn’t the same as that of the red triangle, this does not matter because the movement of the orange non-triangle makes up for it.

    It’s just simple movement of objects people. No real optical illusion going on here.


  28. You people are way to into it.

    Just think, the only other way to make that triangle other than the first way, is the put the orange and green parts like they are.

    The orange has 3 squares until it has another layer thing(the 2 squares ontop of 2 squares).

    The green has 2 squares until is meets another layer. That’s where the gap is.

    **Think of tetris.**

  29. Well, I just tried cuting out the pieces myself, and noticed that you have to draw the large figure first and then draw the little pieces inside. The pieces, as dimensioned int he first figure, do not fit correctly into the large figure. I’m not going ot argue, but if you want proof, just try this:

    Draw a 5 x 13 right triangle.
    Make a 3 x 8 and 5 x 2 right triangle, as well as the yellow and green pieces to specification.
    Try putting the individual pieces you made into the big figure. You will hopefully notice that they don’t fit, wither way you try to put them in.

    And for all the people who just looked at the “levels” “arrangement”, or “number of boxes”, how do explain the fact that both of the large figures are the same dimensions, and made of the same dimensioned pieces, but a chunk is missing? Area doesn’t dissapear…

  30. Does this mean I have to read the comments including math in them?! I saw this on metacafe.com. Check out their illusions. Cool! One of them teaches you how to draw illusions. I tried it, works!

  31. Okay, I may be an idiot, but this isn’t making sense to me. The curved line explanations aren’t relevant to this illusion. There are no curved lines. If the lines were curved we would be able to see it. People wouldn’t be able to construct their own puzzle out of graph paper and arrive with the same missing square.

    I know it has something to do with positive and negative. Like the riddle about the money that was split three ways and when it was returned, they were a dollar short. When it comes to this puzzle though, I don’t get it. Can anyone give a simple clear explanation that doesn’t involve curved lines or stretched triangles?

  32. Wow. All of you people who try to come in here and sound SO smart talking about curved edges and concave and convex lines and whatnot. Instead of trying to be all high and mighty…why dont you make one out of paper and see how it actually works, it took me two minutes to make one (using a straight edge, NO CURVES) and as soon as i had made it the answer became obvious. The fact is, in this situation we arent talking about the area of the triangle, what you need to consider is the size of the edges of each object. If you go and make one out of paper then rearrange them inside the hole you cut it out of you can see that both shapes have the same area. It is interesting, but what happens is, because the larger triangle is 3 high and the smaller is 2 high you have to stretch out the two L-shaped pieces, instead of leaving them lying on top of each other, so that the two of them will equal the length of the large triangle’s bottom side. Its hard to explain exactly how it works, all I can really say is if you still think it has something to do with the triangles being curved or different sizes or something, its none of that, just make one and you will see. Its very easy. And PLEASE people…don’t try to sound so high and mighty all the time, just makes you seem REALLY dumb when you end up so wrong.

  33. Ummm… I kind of understand it. I don’t get all of the “2*5/2=5” or the “3*8/2=12” and the other stuff. And to all of yous math people with the curved lines, maybe the person drawing it made the red and green pieces a bit off by accident.sheesh.

  34. Thank you, Dizee. This is how it works…Rearranging the pieces like that causes the hole because of the two L-shaped pieces. Notice that the long part of the orange one is longer than that of the light green one. The triangles are there to compensate for the space…don’t get me wrong, I’m not talking about area, I only mean that the triangles could not stay in their original positions. Thank you.

  35. For everyone that has tried to explain the convex,concave fact, good job.
    For everyone else that doesn’t believe it’s that simple, sorry, but it is.
    Let me try to shed a little more light. In order for this to be a true, “straight” triangle, any point along the diagonal should have the same slope (remember “rise over run” from geometry?) The whole triangle has a slope of 5/13, but the red one is 3/8. These are not equal, but they necessarily MUST be if the triangle were proper. Additionally, the green triangle has a slope of 2/5 which also isn’t equal.
    If you’ve tried to construct this on graph paper, no doubt you counted the squares the same way and therefore have “introduced” the necessary flaw into the construction. That’s why it works.

  36. I still dont understand how it being a quadralateral makes any difference… but I’m not amazingly clever and I hate geometry, so…

  37. How can you not get this?? The portions are NOT THE SAME for the red and dk.green parts. They cut off some parts. Those parts all equal one more square that fills in the spot. I didn’t use photoshop or anything? I just used ym 6th Grade mind!!

  38. Look at the slope of the two triangles. (For those who don’t know, slope is calculated as rise over run) The slope of the green is 2/5. The slope of the red triangle is 3/8. Now the difference between the two is very small, and in the picture, it is hardly noticable, but those who said that it is a 4 sided object are correct

  39. i saw this puzzle over a year ago, and aven as a 15 year old it was simple to me, i immediatly saw a difference in ratios between the verticle heights and the lengths if the traingles, being 3:8 and 2:5 which must therefore for a 2 lines when connected, not the apparant 1. It is this difference in the angles of the 2 lines, being an outward angle in the bottom “triangle” (not no more), which adds th area of 1 square to the shape, so the area of 1 square can be removed from the bottom.

    Clever, but not clever enough im afraid.

    I even beat my maths teacher at this too

    so thanks

  40. It’s funny to me that some of the people responding “oh it’s so easy, you’re all dumb” are wrong.

    The figures made by the 4 shapes are NOT triangles, they are quadralaterials.

    Make a transparency (as someone earlier suggested) is the easiest way.. and doing it on a computer is best because you can make sure you are 100% accurate.

    Of course, you can do it mathematically, but those that would think to do it that way wouldn’t be suckered in by ‘assuming’ similarity or congruency, would you?

  41. First, if anyone actually reads down this far, like I did… wow, you have too much time on your hands.
    Anyways, It’s been explained right multiple times, but not to the point of where a, say, 10 year old can do it…
    SO. This site explains it perfectly, and illustrates it so you can see it.
    You see, the two full “triangles” aren’t real triangles, BUT THE SMALLER TWO ARE. If you cut it out and put it together you WILL get the same answer. BUT: try it with these demensions and you’ll see the illusion alot better: The top triangle has length 3*5, and the bottom one gets 2*1. The other shapes become 2*1 and 1*1 rectangles. When you transform it, switch the triangles the same way, and put the larger rectangle above the smaller one. You’ll REALLY see the difference.

    Or? Just draw a line from the very top corner of the first big triangle to the left-most corner in photoshop, then zoom in on the point where the two triangles meet untill you are bored.
    If what I said doesn’t make sense… then you must not know what either a triangle, a line, or a rectangle is…

  42. plain and symple trigonometry and geometry puzzle…. puzzle i said?

    as easy as this:

    it is not the same a square area inside 2 by 8 pices (16) than a square area inside a 3 by 5 pices (15)

    If you sum the pices of both non triangles you get 15 but if you rearenge the pices like in the puzzle, you have to fill a 16 pices area, the triangles are there to confuse you, their hypotenuses can make a perfect straight line while variating the remainig space that has to filled eighter with 16 or 15 pices is made by the opposite and adjacent legs of the triangles. So it can be a space made of 5 by 3, or a space made of 8 by 2…

    you just have to forget about the triangles and focus on the remaining space designated by their opposite and adjacent sides…

    Ho Drakon, Ho Megas

  43. Simply put, The rise of the red and green triangles are not the same. Green has a steeper rise than red (saw this in photoshop and with cutouts). If you look carefully you can even see that the bottom and left side of green is 5 by slightly more than 2 and red in the same area is 5 by slightly less than two. Never took geometry but obviously the extra space is to compensate for the greater rise of the green trangle. Try it and you’ll see. Cut out both trangles and simply line up the points. You’ll see that green has a steeper rise than red. Hope this doesn’t sound stupid. I tried.

  44. ooh i know this one! The triangles are not exactly the same shape. The top one has bent edges but they’re really hard to see.

    I saw this on some web page with the proper explanation but I dont remember where… hehe, sorry.

  45. ok woooww it definitley has NOTHING to do with bent sides…in the top triange, if u take out the two ‘L’ shaped objects, u have an empty 5 by 3 rectangle which means the are is 15. In the bottom triangle, if u take out the ‘L’ shaped objects, u have an empty 8 by 2 rectangle which means the area is 16 and there is one more square then the top triangle.

    idk how that works but thats all i could think of

  46. Find the tangent (tan) of an angle A in a right angled triangle with side 1(opp) and 4(adj). The hypotenuse is the side of the triangle opposite the right angle. The opposite side is the side of the triangle opposite A. The adjacent side is the third side. To get Tan(A) divide the length of the opposite side by the adjacent side. So in this triangle, Tan(A) = 1/4.
    A triangle with Tan(A) = 1/4

    For example, the steepness of a road is often shown on road signs as a percentage. If the road surface is at an angle A away from the horizontal, the steepness percentage is 100*Tan(A). What steepness percentage is a road at 45 degrees? How about 63.4 degrees?

    I could give u d answer but dats not as fun !!!

  47. the answer is easy. The 4th column from the left is larger than the other ones. so the area isnt the same for the red one in the top triangle as it is in the bottom one.

  48. *****SPOLIER ALERT*****

    the real trick is in the smaller red triangles go back and count how many squares are in them. you will find the top one has one more

  49. I don’t know if people will read this, but here’s the explanation to why it works when you make your own triange.

    I agree with all the people who say that it has to do with a bent line. That is correct. The people proved this with easy math.

    So what happens when you use all straight lines, you ask. All that happens is that the second triangle is bent outward even more that it would be if you originally started out with a concave line. The difference in area between the triangle with straight lines (the first one) and the triangle with the convex line (the second one) will still amount to one square.

    The reason that the makers don’t start with a straight line for the first triangle is because it makes the second bulge less obvious by spreading it out between the two triangles.

    I hope this clears things up with people saying that they made their own triangle and it worked.

    p.s. And just in case you didn’t get it before, the triangles I am talking about in my explanation all refer to the big triangles, not the two small ones that make up a larger one.

  50. The people laughing like idiots and saying it has nothing to do with bent lines fail to explain why the bent lines EXIST.

    Of COURSE it has to do with the bent lines, otherwise the lines would not be bent in the first place.

  51. uh, öh uudndfnbfkjhdf!!!! I MADE IT FROM PAPER!!! eefeeudyd!!! OMG!! OMFG!! ZOMG!! I CANNOT UNDERSTAND IT!!! EEEEHakkaöaöä!!

  52. the red one is the biggest so u have to move the green one back a space it took me 10 seconds to figure it out and i’m 13

  53. ***Spoiler Alert***

    They have different areas because the hypotenuse (top left line) of both triangles are not actually lines. They are to separate lines which inter sect at an angle just above or below 180 degrees (depending on which triangle). Since the lower figure has a bigger angle it takes up more space and therefore leaves no room for the missing block.

    This is how I know that the two hypotenuses are not actually lines:
    In order for a line to be a line, it has to have the same slope throughout the whole line.

    If Slope is m
    m = Height / Length

    The segment of the red triangle that makes up the supposed top-left line goes up 3 BLOCKS and across 8 BLOCKS so therefore has a slope of “3/8”

    The segment of the turqoise triangle that makes up the supposed top-left line goes up
    2 BLOCKS and over 5 BLOCKS and therefore has a slope of “2/5”

    Since the two line segments have different slopes, they can’t possibly make up a single line segment. Therefore, the two big triangles are not triangles afterall, but are actually quadrillaterals (4 sided polygons).

  54. Wow its easy the area of wich the blocks were placed is diffrent so it messes it up…. well its hard to explaine…. and whats with the pointless math its just a frikin elousin!!!!!!

  55. If you have a curved line that takes up space and then you take away that curvature it doesn’t produce a square out of thin air. The area of one square is either contained or not contained within that curve. Especially if the damn things are not pressing against any other objects and just hanging out on top. I don’t have an answer to the puzzle but that’s my response to the curvature theory. BS!

  56. ohhh my god nr 5 is right all u other smart ass’s diden’t get the point, of the Illusion, im lauging my ass of haha, u think u are so smart yet u are so dum,,,, just look at the link at the 89..

  57. Use the grid as if it was a graph. You would see that the slope of the bigger triangle is 0.375 and the slope of the smaller triangle is .02, which indeed is a big difference. That means both the lines on both the triangles are curved. The first “whole” triangle is concave (going inwards) and that makes the triangle look plausible. The second triangle is slightly convex (curving outwards) which is hard for the human eye to notice. That little difference is actually equal to the missing square’s area. so the curved lines are the whole idea of the illusion.
    If you use a very accurate way to measure this you would see that what I am saying is true.

  58. I’ve figures it out
    The Top triangle goes up by 5 squares
    And ythe bottom triangle is 6 squares but it tries to trick the human mind by adding a rectangle istead of a square but they’re
    5:6 not equal

  59. The area of two triangular pictures is not same.

    Red triangle area in both pictures = 3*8/2 = 12
    Green triangle area in both pictures = 2*5/2 = 5

    Area of two blocks in top picture = 3*5 = 15

    Area of two blocks in bottom picture = 8 + 7 + 1 = 16 (each smal box is considered as 1×1)

    The two pictures look equal but they are not equal. Zoom and look at the 6,3 from bottom left on bottom picture and then compare the same location on top picture.

  60. For those of you who don’t understand geometry, try this:

    Cut out the all shapes TWICE.

    Arrange the shapes like the two sets above then put one whole set on top of the other.

    Are they the same size?

    You should be able to see that the longest side is different in each set.

    This is why, try this:
    Put the red triangle on top of the green triangle so the pointiest parts are together.

    Are they the same angle?

    You should be able to see that the angles are different.

    Now, if you can’t understand that, then we need to introduce an IQ based population cull …

  61. Honestly, this whole thing has nothing to do with curves.
    The curves are just slight errors. The real mathematical phenomenon occurs with the orange and green non-triangles.
    The area of these two pieces will never change. But by moving the bluish triangle to the bottom row and the red triangle to the top, the perimeter of the orange and green non-triangles, combined, is forced to change. If a shape’s perimeter is altered and is not allowed to change area, a gap is formed.
    That explains the gap, but I am still unable to come up with the larger question, “Why aren’t the areas of the two triangles the same?”
    I repeat: There is no trick in the curves, just cut the damn shapes out yourself using a straight edge.

  62. 1. Two Non-Triangle shape dont make any difference. This streching-out thing which has been explained by some posters is pure S***. They are more adept in Tetris rather than Geometry.

    2. As explained earlier, difference is because of Concave and Convex Hypotenuse ( Calling it so for namesake.) It actually IS NOT a single line.

    Mathematical proof has been given in above posts. I'll just make it simple for those who are making it on graph paper.

    > Draw TWO sets of the above image. NOT in single piece as BIG triangle, but with 4 given small figures.(precisely to scale)

    > Arrange set ONE to form top triangle.

    > Arrange set TWO to form bottom triangle.

    > Place the FIRST formed triangle on top of SECOND formed set.


    > Have some belief in yourself. This is not because you made some mistake in cutting. This is there because a DIFFERENCE IS THERE.

    I cannot state this in simpler form than this. If you still haven't got it, Enjoy the mystery and don't try to solve it any further.

  63. I figured this out in a minute, But my dumb ass friends still don’t believe me!!! so they cut out peaces of paper and tried to prove me wrong. Oddly enough because it is so minuet they “proved” me wrong until i told them to fit it into the “triangle” hole they cut it out of.

    Like said before, simple geometry.

  64. it is like some of you retards want to believe that this is ‘magic’.

    those are not actually triangles by definition, so the area of the two using 1/2*b*h is irrelevant. plus they are obviously two different shapes so you wouldn’t be able to use the same formula to calculate area anyways…

    most people should be able to understand that since the two actual triangles don’t have the same angles, the two hypotenuses connected can NOT make a line, but can only make two line segments…

    but for you retards- pick a vertical grid line and look where is crosses the ‘hypotenuse’ of each of the larger ‘triangles’. notice it isn’t at the same horizontal space. ergo, they aren’t the same shapes.

  65. The illusion is simple. The blue triangle is NOT the same in both images. In the original figure the triangle is 5 x 2, while in the bottom figure, it isn’t. How do I know? The slope of the whole big triangle is 5/13. That can’t be reduced, which means that the little blue triangle CANNOT be actually 5 x 2. It’s really about 5 x 1.92, and that small little error, although undetectable to the naked eye, makes up that last open square.

  66. It’s totally as Jiggs up said the interesting part of all is that the RHOMBOID that is formed, is exactly 1 square unit. We use this examples to optimize the areas when designing in architecture, cause it’s a minimum space lost y long distances, then turned into useful square spaces. –Fernando Barragán–Mexico City.

  67. To Do’h: first of all, it should be spelled “D’oh.” Also, as far as I’m concerned, velocity has to do with the rate at which an object moves.

    Comment ten is correct. I don’t really care about it not being a 180 degree angle. What happened is that instead of putting the orange L on top of the green L, it was moved over one square. Since the red triangle is one unit longer than the dark green triangle, it was able to fit.

    Paula: good luck in geometry class. I hate geometry, but if I pass this year, I don’t have to do it again in high school.

  68. look, the triangles have different angles- since tan (red) = 3/8=0,375 and tan (blue) = 2/5=0,4
    that means that both shapes aren’t triangles, but polygons with a blunt corner at the point where red and blue meet.
    so how did we learn to calculate the area of a polygon in primary school? draw a rectangle around it, calculate it’s area and subtrack the area’s of the triangles and rectangles that surround the polygon. (it’s hard to explain for me in english but try to imagine/remember it)

    well, the area of the rectangle is 5*13=65 and the area of the first polygon is therefore 65-33=32 [65-(5*2/5 + 3*8/2 + 8*2)]
    the area of the lower polygon is 65-32=33 [65-(5*2/5 + 3*8/2 + 5*3)]

    the area’s of the coloured shapes remained the same in both polygons, but this illusion makes you think that both supposed big “triangles” have the same area. but in reality they’re two polygons and the area of the last one is 1 square bigger than the first (33-32) and that is created by the empty square you see.

    i didn’t really say any new things, but i’m just trying to make it a little more clear than in #10

  69. Open Photoshop, then start a new project on a perfectly square canvas. Now show your grid guidelines (ctrl+’). You will need to go into the gridline preferences and make the gridlines for every 20%, with 3 subdivisions each. This will display a grid of 15×15 squares.
    The Triangle is 13 squares wide, and 5 high. Draw just one triangle with these dimensions, photoshop will automatically snap to the corners of the grid boxes. You have a perfect triangle. Now realize the dimensions of the orange pattern. It’s supposed to be 5 squares across on the top row, and two squares on the bottom left, occupying 7 of the grid boxes, perfectly. Look at the triangle you drew in photoshop. The top left corner of that orange shape, will not fit cleanly between the red and green triangle, no matter which configuration you have the partitions set. The top left corner of the orange shape would have to meet exactly at a corner of one of the grid squares. But as you can see by the perfect triangle you drew, there is not one point within the hypotenuse that crosses a grid box exactly at the corner. So the Orange segment will not fit as the illusion suggests. In the illusion, the hypotenuse crosses two exact corners of two different grid boxes. This is not so when doing so on a mathematically true grid and perfect triangle. The illustration here is an illusion, the red and green triangles have to cheat in order to cross grid corners. The two triangles have slight curves.

  70. You all saying the bend is the answer are retards, excuse me.

    But you will get the same illusion if you made the figures yourselves witht the most precise accuracy to each other as possible.

    Therefore, the bent line “theory” does not explain it.

  71. OK, now, after reading almost all, and I say almost so that i dont insult someone i think you are ALL WRONG. The key here is not trigonometry, nor in the triangles. Let me explain. I think the answer is this – surface area and circumference are NOT in direct relation. This would mean that objects with the same circumference can have different surface area and the other way around. So – Leave the triangles aside, the difference in the two pictures is not their surface area. Its their circumference. Look closely. If you count the squares, their number is the same in both picures, its just that in the bottom picture the overall circumference of the figure is greater. Again to make i easier I will leave the triangles, they are there for confusion and go down to the numbers.
    In the first picture, we have a rectangle with a surface are of 3×5 = 15
    Now. In the second we have a figure that is not an exact rectagle (look only at the coloured part) but this figure stil has a surface area of 15 (just count the coloured squares). So where is the difference?As i said the difference is in the circumference. the first figure (the rectangle) has a circ. of 2×3 + 2×5 = 16 while the second has 2×8 + 2 = 18. (If you a re wondering why I add 2, I will explain. I calculated the circ. of the rectangle with sides – 2 and 8. then subtracted the non coloured bit, which was 1 (remember we are talking about sides, now squares here), but had to add 3, which is the number of coloured sides, all in all -1 + 3 therefore +2. I hope you got it. To sum up. The key in this is the question. And almost all you had it WRONG. the two coloured figures actualy have THE SAME SURFACE AREA. Question is why do two different circumference-s belong two figures with same surface areas? Now proving that in generalall cases is much tougher, but you can make it look easier with a simple exercise. Imagine a cube with a side =4. Now this has a circ = 16 and a surface area again = 16. now draw a rectagnle with a side A = 2, and B = 6. What do you notice. They have the same circ.-s = 16, but the surface area of the second is 2×6 = 12 which is NOT 16. There you go. All non linear conspiracy theories are refuted.

  72. I got it in less than 3 minutes. I still don’t understand why everyone is talking about trigonometry and that stuff. It’s simple! It’s just a different arrangement of the shapes! Move the orange shape to the side and (a la “tetris”) watch it fall and form the hole. I would tell you more, but I don’t want to feed the beast.

  73. It’s not as hard what everyone seems to think. There’s no “Line Bend” that creates enough area for the “hole”. The fact is, the “hole” is not part of the area for the triangle. you can’t use a trigonometric equation on the second “triangle” because it’s not a triangle. By moving the individual colored pieces, one can simply change the shape, and while it appears to be a triangle, it isn’t. The pices still have the same area, and this can be easily replicated: just make your pices [either on paint/photoshop, or with some paper] and rearrange them like on the picture. No “magic line bend” required.

  74. The Hypotinuse is NOT a straight line in either image. it bows slightly outward in one and inward in the other. The gridlines distract your eye from noticing it.

  75. I have not read all comments so forgive me if this is being restated. The two trinagle are not the same proportion. They have a slightly differing slope, but it is close enough to trick the eye.
    The red traigle is 3/8 and the Greaan is 2/5 or 3/7.5. That .5 differsnce allows for the extra square to be created when the pieces are moved around. Thats all there is too it

  76. I knew in about 5 min that the squares are not perfect squares in the grid.It’s the only way to explain how it’s possible to “gain” an extra space. Makes sense now as I looked down on hypotenuses they’re not parallel.

  77. just look at the slopes of the hypotenuse of the red and cyan triangles. they’re not the same, so it isnt a continuous straight line

  78. Look at the orange shape and the lime shape. The orange shape has two squares on its head and three squares on its tail. The lime shape has three squares on it’s head and two squares on its tail.

    On the top, the head of one shape lines up with the tail of the other. That makes a rectangle that is three squares high and five squares long.

    On the bottom, the orange shape is moved so that the tails touch. The tails don’t match though. So, that creates the gap. It also changes the rectangle to two squares high and eight squares long.

    The red triangle is one square higher and three squares longer than the green triangle. So, when you move the red one up, it makes up for the decreased height of the two non-triangles in the first image, like adding 1 and -1. It also stretches to cover their increased length as well.

    The green triangle is only two high and therefore matches up with the second rectangle. Since it is three squared shorter than the red, it ends up stopping where the red triangle originally stopped even though it is attached to a rectangle that is three squares longer. It’s like adding 3 and -3.

    There is no net change in the overall shape of the two triangles other than the gap.

  79. I laugh at the people talking about curved lines and such. Draw it out with perfectly straight lines and you will get the same result. It’s pretty easy to figure out what the problem is as other people have pointed out. The red triangle has a base length of 8 squares and a height of 3. The green triangle has a length of 5 squares and is set 3 up from the bottom. That makes the space with the other two objects have an area of 15sq (length x width L is the base of the green;5, and W is the height of the red;3. 5×3=15). When you switch the triangles you are changing that space from 15sq to 16 sq. The length becomes 8 (base of the red triangle) and the width becomes 2 (height of the green triangle)so the area left to fill is 8×2=16sq. You can’t take two pieces that fit together and have an area of 15 and move them to complete an area of 16.

  80. @Anonymous comment 120
    do you know the difference between curves and angles? Apparently not. Let me give you the definition of a curve.
    curve (kʉrv)

    Archaic curved

    Etymology: L curvus, bent: see crown


    1. a line having no straight part; bend having no angular partI will admit that after going back and looking closer at the first 13×5 triangle drawn on graph paper, the 5×3 point on the hypotenuse is slightly below the grid line. So it’s an angle and not a curve. Either way it still has to do with the area of the object changing when you rearrange them. I’ll even quote from the site, “The top figure has an area of 32 square units. The bottom figure, including the empty square, has an area of 33 square units.” Which is because it’s not a triangle.

  81. Even if you don’t get the math–it doesn’t matter. The point of optical illusions is to point out how your brain tricks you into “seeing” something a certain way, regardless of the physical reality of the actual thing you are seeing. That’s what is really the mystery here–the way our brains work!

  82. Take a blank sheet of paper and line up the edge with hypotenuse of the red triangle. Watch as the hypotenuse of the green triangle magically jumps off of the edge of the paper!



  84. i cut it out of paper. it still gives an extra square, so it is not an illusion and it is not because of bended lines. i can’t explain it, but i am really bad at geometry

  85. This is a very interesting puzzle. I remember finding it and figuring it out when I was in 9th grade, but it took me a good while. Very intriguing. I think people are getting hung up on terminology here. There are no “bent” lines per se. All individual shapes in the top and bottom figure are identical and drawn with straight lines. The grid is not flawed either. If you add up the areas of the individual shapes (in either picture) you get 32. If you take the area of the top figure as a triangle you get 32.5 and for the bottom figure 31.5. Neither is correct, so the obvious answer, the large figures are not triangles, but then you muse find out why. The slope of the small red triangle is not identical to the slope of the small green triangle. Therefore, the total figure is not a triangle, but a quadrilateral because it’s “hypotenuse” has a slight angle change at the meeting of the red and green triangles. I don’t want to say a bend because of terminology again. All lines are straight. But the “hypotenuse” of the large figures is really two separate lines that are convex in one figure and concave in the other, accounting for the “missing” unit.

    If this does not make sense to you, may I propose something. Get a piece of graph paper and draw the red (8×3) triangle and green (5×2) triangle on top of each other. To make it even more pronounced you can double or triple all distances, as that does not change the angles. Draw an 24×9 triangle on top of a 15×6 triangle using a ruler. You should very clearly see that their hypotenuses are at different angles.

    Warning, here goes a rant. Now what I find amusing is the people who laugh and say how this problem is so easy, how can everyone not see it? You amuse me. Especially when you bring up, my little 13 year old mind can see this easily! Oh you. If your solution is that the shapes were moved, all I can say is, REALLY?! The rest of us didn’t notice the shapes were moved around! Thank you for enlightening us so!! Now please go back to your tinker toys and lego while the rest of us move forward with our lives by using our brains. If your answer says anything about the two triangles in the first figure producing a 5×3 cavity (15 area) while the two in the second figure they produce an 8×2 cavity (16 area) or some mention of tetris, probably followed by a comment about how you don’t believe everyone can’t see this right away, let me explain something to you. Not only are you too dense to be able to solve the problem, you are too slow to even understand that there is a problem. And that, my friends, is far worse. In the illusion, a unit of area appears to have disappeared. No matter how much you try, you cannot explain a change in area by moving things around. Its like I shuffled a deck of 52 cards and ended up with a deck of 51 cards. And your answer is, well you shuffled it. Ok. Yes I did. But that’s not the problem. Or I moved a table across the room, and it turned into a desk. Your answer is, well you moved it. That’s not the problem. There is an illusion here, and you are not even seeing the illusion, much less the explanation to it. So if you cannot even understand the problem, don’t try to act like everyone else is stupid for not seeing the answer. Understand the problem before you even try to solve it, please.


  86. this is not an exercise in plane geometry as much as an exercise in being able to explain the answer coherently. one person didn’t know the difference between circumference and perimeter, and another the difference between bent and curved. the illusion is that the large “hypotenuse” is a straight line, which it is not. the two small angles of the triangles aren’t equal because 8:3 and 5:2 aren’t equal ratios; therefore when you reverse the triangles the “hypotenuse” bends outward instead of inward. the increase in total volume of the large triangle is enough to accomodate the extra square. this can be proven by simply adding the four individual areas.

  87. somthing u dont notis is the tryangles are bent diffrently but 1 thing id like to say is that i cut out the shapes out of a peice of paper and re aranged them and got the same illusions. savy? (savy means ok)


  88. oh and 1 more thing the bend is just making people think the wrong thing (only the tryangles are bent) if u cut out your own shapes like i did u can see for yourself

  89. If you take the smaller triangles and do a little trigonometry on them, you will see they are not similar. The smallest angle on the small triangle is 21.8 degrees, the big (red) triangle has an angle of 20.6 degrees. I tried roughly sketching this out on graph paper, and they appeared similar, but a difference of 1.2 degrees is imperceptible.

    If you take the top shape and kind of fold it over on the bottom shape, so that the green triangles and the red triangles make a 2×5 rectangle and a 3×8 rectangle, respectively, then you can see this more clearly. You get a 5×13 rectangle with one piece missing, because the area of 5×13 is 65 and the areas of all the colored shapes only adds up to 64.

    There is no bending, just dissimilar triangles.

  90. The “whole” triangle is not REAL triangle, it’s just a look-alike.

    The red&green triangles appear to be similiar (to have equal angles), but actually their angles differ. If you assume all the sizes of the shapes (which form the whole picture) are determined exactly by the number of squares they occupy, you can easily calculate the angles of the red&green triangles, using trigonometry.

    This is a mathematical proof that the hypotenuse of the top (whole) triangle is not actually a straight line, which means that this object is actually not a triangle, which means that its area cannot be calculated (accurately) with the “right-triangle-formula”, but rather “the-sum-of-its-parts” approach should be used.

    The conundrum here is that, the point where the red&green triangles meet appears to be exactly on top of the grid, which gives the erroneous impression that their sides are well-defined. In reallity, this picture is ambiguous. If the grid represents the sizes accurately, then the sides of the red&green triangles can’t be measured with whole numbers (integers), but irrational numbers should be used instead.

    So… my conclusion is that:
    1) The “whole” triangle is not REALLY triangle (its hypotenuse isn’t straight line)
    2) If you get a grid of squares and draw a STRAIGHT line in such a way that it forms the hypotenuse of a right triangle with sides 5 and 13, you’ll notice that this line doesn’t go through any of the grid intersection-points (it will come close, but not exactly over where the grid lines intersect)
    3)This is one EXCELLENT visual illusion.

  91. [i]Anonymous says:
    February 23, 2009 at 10:36 pm
    OK, now, after reading almost all, and I say almost so that i dont insult someone i think you are ALL WRONG. The key here is not trigonometry, nor in the triangles. Let me explain. I think the answer is this – surface area and circumference are NOT in direct relation. This would mean that objects with the same circumference can have different surface area and the other way around. So – Leave the triangles aside, the difference in the two pictures is not their surface area. Its their circumference. Look closely. If you count the squares, their number is the same in both picures, its just that in the bottom picture the overall circumference of the figure is greater. Again to make i easier I will leave the triangles, they are there for confusion and go down to the numbers.
    In the first picture, we have a rectangle with a surface are of 3×5 = 15
    Now. In the second we have a figure that is not an exact rectagle (look only at the coloured part) but this figure stil has a surface area of 15 (just count the coloured squares). So where is the difference?As i said the difference is in the circumference. the first figure (the rectangle) has a circ. of 2×3 + 2×5 = 16 while the second has 2×8 + 2 = 18. (If you a re wondering why I add 2, I will explain. I calculated the circ. of the rectangle with sides – 2 and 8. then subtracted the non coloured bit, which was 1 (remember we are talking about sides, now squares here), but had to add 3, which is the number of coloured sides, all in all -1 + 3 therefore +2. I hope you got it. To sum up. The key in this is the question. And almost all you had it WRONG. the two coloured figures actualy have THE SAME SURFACE AREA. Question is why do two different circumference-s belong two figures with same surface areas? Now proving that in generalall cases is much tougher, but you can make it look easier with a simple exercise. Imagine a cube with a side =4. Now this has a circ = 16 and a surface area again = 16. now draw a rectagnle with a side A = 2, and B = 6. What do you notice. They have the same circ.-s = 16, but the surface area of the second is 2×6 = 12 which is NOT 16. There you go. All non linear conspiracy theories are refuted.[/i]

    So, dude… if you can’t figure out why the apparently straight line isn’t really straight, you can pick your favorite ruler and see this for yourself. Or maybe they’re not that straight anymore? It’s a fact, not a theory. With trigonometry, you can (on theory) see exactly which portions of the elements form the missing square.

  92. long story short, anyone smart enough to recognise the problem already knows the answer. two of the same shapes cant have a different area… so whatever cant be true, isn’t. the shapes are either different, or the area is the same. in this case the shapes are different. END OF STORY.

    but ill keep talking anyways

    if simple theory doesnt work on you you can do as one person did somewhere above and prove it with simple geometry. he figured it out by calculating all dimensions looking for the inconsistency, and found it.. a different angle on the two triangle.

    you cannot argue that you built the pieces and it still works, because that is 100% correct, you will accomplish the exact same effect. the illusion here isnt how it is drawn, its just taking a small area, hiding it over a long distance or focusing it in a small space, the small angle is just real hard to see with the eye, and will be only hidden better by trying to tape little pieces of cut out paper together. now if you cut these out v ery accurately, and large enough to work with, and very accurately traces the perimiter, rearanged them as shown, you would see a gap totaling the same area develop, spread out along what is a bent line. basically a waste of time thought, all you did was copy what was already shown on your screen.

    some of you are making this way way too complicated, others are too stupid to understand the problem. the trick to this isnt math, geometry, trig, its simple theory. the area an object cannot dissapear by rearranging pieces of it, but the resulting shape can change. so the shape of the two triangles MUST be different. and it is.

    the missing area is hidden along the length of the hypotenuse by the different angle created.

  93. This is quite simple to figure out. The slope of the hypotenuse of the two triangles are not equal. The smaller green triangle’s hypotenuse has a slope of 2/5 while the larger red triangle has a slope of 3/8. Since these slopes are not equal, then the hypotenuse of the assembled triangle is not a straight line.

    In the top figure the long hypotenuse is actually concave (bowing inward) while in the bottom figure the hypotenuse is convex (bowing outward) making the contained area of the lower triangle greater than that of the top.

  94. The guy talking about circumfrences is a retard. the line IS NOT STRAIGHT! the red triangle and the green triangle have DIFFERENT ANGLES. This makes for the big shape being created to NOT be a triangle, it only appears to be a triangle.

  95. The longest line in both shapes are not straight, they are bent. The ankle of the red shape is not the same as the green one. When we want to know the size of the shape we draw a rectangle and calculate its surface area (of the first shape). That is 13*5=65. Then we calculate the first shape without dividing 65 by 2. First we calculate the green triangle. 5*2\2=10\2=5. The red triangle is 8*3\2=24\2=12. The green shape + the red shape= 5+12=17. Next we calculate the rectangle made out of the orange and green shapes. That equals 5*3=15. 17+15=32. So 65 – 32=33. That means the first shape’s size is 32 and the second shape’s size is 33. That extra one in the second shape gave it enough room to put one extra square.

  96. Yes most of the comments are right.
    I actually calculated the area that is created by the bent hypotenuse. The first picture has 0.5 area less than a straight hypotenuse would have had. The second picture has 0.5 more. add those two up. getting 1 whole box

  97. oh, i see it now
    the spare square comes from the orange part cuz it doesn’t fit horizontally to the green part.
    and also the first ‘triangle-like’ shape is not a triangle

  98. The second triangle is greater than the first few millimeters, 1 millimeter to the left of 1 millimeter and one millimeter from the top right, and because of this that space remains empty, is tested by me, porblema solved!

  99. ★★★★★★★★★★★
    1. The hypotenuse is never straight. Some of you that say it is, can try with a piece of paper and find out it’s impossible, because a tiny bit of the orange piece will stick out of the hyp. anyhow.
    2. Just grab a ruler or anything perfectly straight, place it on the hypotenuse on your monitor. (They are never straight. The thin slice of the area missing (in the first triangle) or in excess(in the second), answers for where the empty area went.
    Do the maths on your own, but i do believe we need no numbers and calculations for this.

  100. That’s a pretty awesome puzzle. We immediately assume it’s a right triangle, when it actually isn’t; very tricky.

  101. Hi,

    I didn’t read all the comments but I saw that some of you have found the answer. Some of you tried to cut the figures in a paper and said that this is true. Well, they forgot one IMPORTANT thing: they forgot to put it in a canvas, then they should have seen that by changing the position of the figures, it didn’t fit anymore in the canvas. The extra is very small, it corresponds to 1 square distributed equally around the whole figure.

    Now to the scientific explanation: il you move some figures in a different way, it can’t gain or loss size (English is not my mother tongue so I don’t know if I’m clear enough), those figures will always have the same size, unless it is ice figure : ) But we you keep saying that we see a square in the example. Well then that means that the figures in our case have shrinked to fit the canvas. That’s why people who tried this in a drawing software like photoshop realise that the seconf figure can not fit in the same canvas.

    Sorry, nothing grows or shrinks when you move then, they always keep the same size.

  102. spoiler:

    the first assumption that most people make is that all of the parts put together as they are make up a triangle, this is not true.

    the smaller green triangle is 2 x 5 (40% slope)
    the larger red triangle is shallower at 3 x 8 (37.5% slope)

    on the top set, the green and red triangles create a concave(inward sloping) surface, and when switched they create a convex(outward sloping) surface giving you the size difference.

    1. MuntainDW is right , this curvature that he spoted does not be so clear due to the thick black lines that are used .You can see that litle diference if you count from left to right five boxes and then two up , if you compere the up (red) and the down (green) you will see tha the coloured aera is different , before and after the critical dot .
      The key of the illusion is that the diagonal lines are not straight.

    2. Is not a triangle, 2:5 is not 3:8, wrong, check you, the hole always exist, only need paint a triangle, impossible partition of parts with this.

  103. This is nothing but a scam, it’s full bullshit, trie to draw it with 3 straight outer lines and it’s impossible to get the same area of the triangles. The angle of the slopes differ so little that you almost can’t see it on the drawing. It’s maybe even less than 0.05 degrees!

    But the trick is quite cool :)

  104. I’ve finally figured it out….

    A bit difficult to explain…
    The “issue” is that with the large red triangle (top picture), the drawing seems to put each of its corners into the corners of a square, in the graph.
    But one of the corners actually does not go into one of the corners of a square (the fact of which is disguised, by using the black thick line).

    And so the trick is… when a new picture is drawn, it is drawn in a way, where the red triangle is a full triangle, with all corners fitting exactly into the corners of the squares – this is not representative of the triangle which is in the top picture. The diagonal line should be shorter)

    If you print and cut out the shapes on paper, things might become clearer.

  105. It’s quite simple, actually.
    Green triangle: 5×2
    Red triangle: 8×3
    When you move the red on top, and the green to the bottom, you get 1 extra square of height and 3 squares less length.
    To account for the height and length difference, you move one of the square~ish blocks down, giving you 1 sqr less in height, and 3 sqr extra in length.
    Now, these blocks aren’t a perfect fit in this position, due to the ‘neck’ being 3 sqr and the base being 2.
    Thus, you get that empty square in the bottom.
    While the total triangle keep the same shape.

    1. you are wrong. these figures are not a triangle. they seems to be a triangle but in fact they are 2 different quadrangle. (this is an illusion, remember?)

  106. its the triangles if you cut the red triangle right in the middle to match the green one you’ll see that the green one is bigger. By switching them around you can create a steeper slope of the line or a flatter slope. This is how the extra block is made. Since the green triangle was on top of the red it caused it to take more space because of its steepness and caused no extra space. And when you switch them around there is extra space.

  107. I don’t know if it matters, but the slopes on the red and green triangles are slightly different. The green one is 2/5 and the red one is 3/8…

  108. The hole come from the rearrangement between partion green and partion orange.they both have 5 cubes,but partion green has 3 cubes and partion orange has only 2 so when rearranged it creates a hole.

  109. this is too easy to solve
    first picture look at coordinate 6,10.9
    second picture look at coordinate 6,3 – notice the small tip difference
    the difference in that point makes the first picture coordinate = 6,10.9, instead of 6,11

    for the first picture, when the y point is 11, you’ve to move x to 6.5 > so that extra 0.5 extends the length of red triangle

  110. The colored shapes are the same in the two cases. What is not obvious, and this is explained by MuntainDW, is the shape that the parts make. The two shapes are not triangles, OK? It’s an illusion, that’s all. The two shapes are not right-angled triangles because the hypotenuse is not a straight line. To be a straight line, the red and green triangles have to have the same left side (sharp) angle. For that, the tangent (or slope) can be verified:
    – RED triangle tan=3/8=0.375
    – GREEN triangle tan=2/5=0.4
    So they determine a concave (inward) “hypotenuse” in the first case and a convex (outward) “hypotenuse” in the second case, the area being slightly larger than the first.

  111. Both of them are not triangle. they are polygon.
    This case proves the Gestalt Theory.
    People try to get things in a whole by form

  112. If QI is right, then the edge of the green and red triangle is actually sloping. A French magician invented it – can’t remember which one, though.

  113. The curves of the triangles are not exactly the same. They slope inwards on the top, but outwards on the bottom. You can tell at the red tip on the bottom picture.

  114. lol, my girlfriend ask me today about it, and i make my own proven on corelDraw.. and the answer is, the hypotenuse is not the straight.. here my proven [img]http://yfrog.com/npimposibletrianglesolvej[/img]

  115. The key to the puzzle is the fact that neither of the 13×5 “triangles” is truly a triangle, because what would be the hypotenuse is bent. In other words, the hypotenuse does not maintain a consistent slope, even though it may appear that way to the human eye. A true 13 × 5 triangle cannot be created from the given component parts.

    The four figures (the yellow, red, blue and green shapes) total 32 units of area, but the triangles are 13 wide and 5 tall, so it seems, that the area should be S=\frac{13\cdot5}{2}=32.5 units. But the blue triangle has a ratio of 5:2 (=2.5:1), while the red triangle has the ratio 8:3 (≈2.667:1), and these are not the same ratio. So the apparent combined hypotenuse in each figure is actually bent.

    The amount of bending is around 1/28th of a unit (1.245364267°), which is difficult to see on the diagram of this puzzle. Note the grid point where the red and blue hypotenuses meet, and compare it to the same point on the other figure; the edge is slightly over or under the mark. Overlaying the hypotenuses from both figures results in a very thin parallelogram with the area of exactly one grid square, the same area “missing” from the second figure.

    According to Martin Gardner,[1]the puzzle was invented by a New York City amateur magician Paul Curry in 1953. The principle of a dissection paradox has however been known since the 1860s.

    The integer dimensions of the parts of the puzzle (2, 3, 5, 8, 13) are successive Fibonacci numbers. Many other geometric dissection puzzles are based on a few simple properties of the famous Fibonacci sequence.[2]

    1. In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is that rate of increase. In simple terms, the variation in space of any quantity can be represented (e.g. graphically) by a slope. The gradient represents the steepness and direction of that slope.

      The green triangle has dimensions 2 x 5 and gradient 2 / 5 = 0.4

      The red triangle has dimensions 3 x 8 and gradient 3 / 8 = 0.375

      Hence the gradient of the green triangle is greater than that of the red triangle.

      An exaggerated outline might look like….

      In summary the missing square in the bottom triangle is made up for by the fact that it’s hypotenuse bends out where as for the top triangle it bends in.

  116. Thank you Mysteryman and all the others who gave this one away to me. I just COULDN’T figure it out, and it was driving me nuts, causing me to question my sanity and the basic rules of geometry. Now I can sleep peacefully again.

    1. Illusionman is indeed smart, too smart I guess for ordinary people like the majority including me. What most people need to learn from a Math teacher, and from any other subject for that matter, is to make the solution easy to comprehend, and in this case, visualize.

  117. Hrm.. interesting ideas, but, i just did this with graph paper, and it works. there is a new square out of no where.

  118. I think most of your explanations are wrong
    as i painted them on a paper and they are identical .
    There is no missing parts or gimmicks :o

    1. Yes, I redrew this in Adobe illustrator and all the pieces are the same size. I can’t figure this one out.

  119. The original text lied, mysteryman is right they are not the same size. I reopened the image in photoshop and using the pen tool drew straight lines along the hypotenuse. It is slightly different in each drawing, bowed out slightly and bowed in slightly, giving the extra space for the ‘hole’ I see it, understand the animation, but it still doesn’t make sense somehow.

  120. The acute angles in the red and green triangles are marked as equal when they are — in fact — not. If the large triangle actually existed, the two angles would be the same, but they’re not and it isn’t … It’s an optical illusion. A good one, but still an illusion.

    Bill (the author of a math book)

  121. Get millimeter paper and draw the exact triangle. Then you will see the difference when you start dividing triangles with ruler. The Big one doe not match the squared paper lines.

  122. i don’t get it all this drama and math at the comments, they got the pieces and moves around, same pieces, different positions equal different result. am i really dumb or really smart???

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