# Mathematic Triangles Illusion

By on January 4, 2007, with 52 Comments

I was so thrilled when I found this optical illusion. It’s ancestor (see here), was one of the hardest optical illusions for me to understand. I had really much trouble with the original, but after few days I finally got it! When you understand the principle, you understand all the variations, like the one you see posted below! It’s pretty cool, don’t you think? Try figuring out the solution. Share it with your friends… it’ll give them some interesting moments. I love math as well, so it’s kinda “emotional” for me, if you know what I mean ;) You can also try and solve another, rather easy math optical illusion, I posted it in my early days. Enjoy!

52 Responses
1. Coqui says:

Hey it’s so cool! Love your site and widget!
=D
Coqui

2. Anonymous says:

In fact this aren’t the same tiles,
thats the only explanatiion

3. MED says:

very cool.

quite simple tho. the shapes in the second diagram are not exactly the same as in the first.

if you take the shapes from the first digram and rearrange them as shown in the second diagram you will find a very narrow gap between the two sets of triangles, in a kind of ellipse shape.

Either cut and paste, draw ur own or just use trig :-)

• zoro says:

i think you just cut your figures wrong :D

4. tgc says:

i’m lost on this

5. SWEEET DUDE says:

you just have to count the left edge of the purple tile and the top edge of the green tile and you see how it adds up to 13…thus the 5*13

6. mitchell... says:

The Illusions Is… HEY! There’s My Hershy Kiss From Last Weekend! But..It’s Vanilla? Sombody Tell me Why…

7. spoon says:

they are the same shapes the extra “square” comes from a sliver down the middle diagonal of the second figure that you can’t see due to the the thickness of the black lines.

if you don’t believe me…see for yourself. take graph paper and cut out the first figure. then put the pieces together to make it look like figure 2. you’ll see that there is some space down the middle diagonal that causes the area of the rectangle itself to increase while the total area of the peices will of course remain the same.

i’m too mathy for my own good sometimes.

8. a bigger wedgie says:

Yeah, yeah, like Spoon said..
It’s the rearrangement of the diagonal lines that creates the extra area.

9. Anonymous says:

The small diagonal part in the middle of the 2nd drawing is 3 blocks by 1 block, so the diagonal parts of the purple and the pink figures are 2 blocks, so they should be 6 blocks long. So in real the total diagonal cannot be a straight line.

10. JaqueChirac says:

Chuck Norris can make each of them add up to 66.

At first guess:
the area of the yellow+blue triangle =5*13/2=32.5
The area of the yellow triangle=8*3/2=12
The area of the pink stuff=(3+5)*5/2=20.
Therefore 12+20=32.5
Therefore 32=32.5
Now we all know that is not true (thickness of the line has no bearing on this).
Since yellow and pink are basic forms and yellow+pink is composed of the two, the latter is not a geometric form. Therefore there is no yellow+pink triangle (the “diagonal line” is not really a diagonal nor is it a line, but two rather~180 degrees arranged lines).

To test this use sinus. You are supposed to get the same value for the left yellow angle and, after splitting the pink stuff into a 3*5 rectangle and a triangle, the left pinky angle. They are different 0.35 versus 0.37.

There. I always hated wasting my time with math. Still do :)

12. Anonymous says:

the line down the center of the rectangle is curved. Take a piece of paper and try to match the two corners without covering the line. Its impossible.

13. Anonymous says:

i actually took the time to figure this one out, i think, i cut it out on graph paper and it works, its true, but i noticed when i measured that the rectangle has one less block ( i measured in blocks) on the edges on the inside than the square does, im 12 so this is my logic it might be wrong tho

14. Anonymous says:

No more math!!!!!!!!!!!!! I had to do this at school It scares me. Math not the illusion

15. Joe says:

I see folks have figured it out, but just so it can be easily seen with just this diagram, look at the slopes. The slope between the blue and red shapes is 5/2, while the one between the yellow and green ones is -3/8. Turn the first one and it becomes -2/5, which is close but not the same. The last slope is -5/13, different from both the others. Maybe it won’t help visualize, but it adds up.

16. Ashton says:

Like Spoon said, it is along the diagonal. I’ve done this with grid paper, and if you line up the pieces, there is the slightest bit on the edge wchich would line up to make the extra square.

-Ashton

17. Anonymous says:

yea the second one doesn’t have the same exact shapes. In order for the two big triangles (blue + green and red + yellow) to make a rectangle, they have to be similar to the small yellow and green triangles. but since 5/13 (sides of big triangle) isn’t equal to 3/8, they aren’t similar.

18. tut 7666 says:

I counted the blocks in the square and there are 65 blocks. In the rectangle I counted 65 blocks also.

19. Anonymous says:

You don’t have to be very mathy to figure this one out. Just look at the shapes. They’re the same as in the first picture, just rearranged, yet you get a different volume. Pretty cool.

20. scott says:

Hooray, trig! 3×1 overlap of yellow and green not similar to 8×3 whole, so they don’t fit together. Now my brain doesn’t hurt anymore!

21. Anonymous says:

a square has greater area than any other shape. thus the difference in number of squares.

22. Anonymous says:

a square has a greater area than any other shape thus the difference in the number of squares.

23. Anonymous says:

Yeah, I agree with MED. That makes sense because it is making the triangle longer. But at first glance this seemed impossible!

24. dee says:

Comment No. 9 is right: imagine the edge to edge cut of a tile of 8 by 3 – green and yellow: factor 2.6666 and a cut of a tile of 5 by 2 – cutting segment between purple and violet: factor 2.5. The angle definetely cannot be the same -> the tiles (e.g. red and green) cannot fit, the gap is hidden by a thicker line that bends a bit.

25. Anonymous says:

The slopes of the triangle and the “trapazoid” are different

26. MED says:

OMG. I aint no scientist but here are some of the daftest comments I have yet to see posted on this site….

No.23 who cut them out but somehow managed to get the new shape to be only 12 squares long :-S

No.28 who thinks 8×8 = 65… d-oh!

but the top prize goes too….

No.31/2 who thinks that shapes can magically change their area without changing their sizes…. hmmm

Well done to everyone else tho, this was a bit of a stinker lol.

27. Anonymous says:

fun one. you can use the slopes to find the extra space. first image slope 3/5 does not equal 5/13 in the second. but it is close which is why it still looks right.

28. Anonymous says:

dats kew

29. Anonymous says:

the diagonal on the right rectangle is not a true diagonal (i.e. it is not a straight line segment).

30. Cornopaez says:

It’s very simple. The two diagonal lines are not exactly the same and don’t have the same ratio. If you look closely, you can see the difference.

31. silver says:

hi! love ur site, but aren’t these just moved around? u can move them back correctly!

32. Anonymous says:

I think it’s because, in the places where a diagonal line cuts through a square, the squares are not cut exactly in half.

all you have to do is this, take tyhe yellow triangle on the left and slide up 3 squares so its the same on the right, now notice the lefis 3X6 while the right is 3X5 thats the easiest way to see the slight difference in slope angle without going into trig.

34. Anonymous says:

its all to do with the parallel postulate…derrrr!

35. the technogirl says:

triangles.

36. Anonymous says:

The surface area of the second one is more that the first. by rearranging the blocks you can make change the surface area to be greater using the same blocks.

37. Michelle says:

noooooooooo math it burns it burns!!!

38. Anonymous says:

use PEDMAS kids…

39. Pavel M says:

The diaganal line on the left picture is not actually a straight line. You can calculate tangents of right angles of Blue and Yellow figures: 3/8(0.375) and 2/5(0.4) accordingly. So the angles are different and diaganal is not a straight line. It’s just an illusion ;)

Thanks! It was fun to solve it :)

40. Pavel M says:

small typo: *The diagabal line on the RIGHT picture.
Sorry.

41. moroccan says:

the square are not squares !! if you have a form of square the result is lower than the other form…look to the squares of the whole table you will see it !

42. Alexander Riverou says:

Maybe all of you already know this,

the explanation is that the slope of the triangle objects in the two figures are not the same, so, that little difference results in an extra square measure unit in the second figure,

in other words, the objects forming the two figures are not same, they are bigger in the second figure.

43. Carlos says:

Sure, the slopes are not the same by pitagoras, so sifferent figures will not align as on the picture.

44. Andrew says:

Also, the thickness of the lines is important in that it HIDES the problem. By being extra thick, it covers up the whole that would be created if you actually cut the pieces from the first and tried to make the second

45. Lolzater=] says:

We did this illusion at school and i said i knew how it wrkz thanks to this site, i told the class all about the website too! ;]

46. Lolzater=] says:

it doesnt wrk because there’s a gap left over when u move the shapes around, equal to 1 square in that shape. It’s like a fake illusion… i also showed the class the impossible objects catagorie and the were FLABBAGASTED [if that's a wrd ;]]

47. crazychik says:

they are the same shapes, the only difference is that in the second image the pink shape is measured on the bottom side while on the first image it was measured on the top. The bottom line is longer that the top.

48. gki says:

well the green and yellow triangles are placed in different places making it wider

49. anonymous says:

just eat your oatmeal and accept that surface area changes!

50. Edmo says:

Quite simple really…..
looking at both the green and yellow triangles…
the hypotinuse in the first image (8×8) is converse, the hypotinuse on the second image (5×13) is convex, which gives you the extra square…. SIMPLES.!

51. Izzy says:

It’s the green and yellow triangles that are about 2% larger in the second picture. The illusion in the second picture is that the hypotenuses of each triangle are a straight line, whereas each actually “bulge” in the middle.

The slope of each hypotenuse changes, so technically, it’s not even a triangle, but now a 4 sided shape. Looking at the diagonal in the second picture, you can see the line goes from a slope of 2:5, to 1:3, back to 2:5. Imperceptible to the eye unless you look really close. The extra width of thick black lines is also used to hide some of this angle change too.