By James Dean on February 24, 2008, with 88 Comments

Hy people! I have just received this image from my brother via gmail. We had very similar mathematical optical illusion at the very start of this website, and I find this kind of illusions hardest to solve. Maybe this should belong to some new category called “Brain Teasers” or “Mighty Puzzles“, still we can probably pull this one of as part of the illusion site as well. So, the question follows: if all the shapes that form these two rectangles below, are of the same shape and areas, how come the overall surface of the second is bigger? Where is the mistake? How do you explain this one?

love it but seen it before… sorry

can someone tell me where the mistake is? i’m curious!

One i can’t solve. Darn!!!!!!!!!!!

The problem is that the large diagonal in the 5×13 is not really a straight line. You can’t really line up the hypotenuse of the 3×8 right triangles with that of the 2×5 right triangles involved in the creation of the quadrangles.

The problem is that the 8×8 uses the longest side of the leftl quadrilateral and the shortest side of the right quadrilateral. on the 5×13 it uses the longest side on both quadrilaterals. btw important 11

I think I see the “mistake” but I’m not sure how to explain…

the solution is in the angles

Complementing Bob’s comment, the shapes of the 8×3 triangles of the two rectangles are not identical. The hypotenuses of that “triangles”, in the second rectangle, was forcelly distorced. In fact, these shapes are not triangles.

No~~ Google ruined the illusion. If you make the picture smaller (which google did), you can see where the problem is. (It’s the abnormality.)

Bob’s right. On the rectangle the hypotenuses don’t make a straight line. What’s more, the right angled triangle doesn’t fit Pythagoras’ theorem.

http://i201.photobucket.com/albums/aa2/franz9265/geometry.jpg

The image explains what’s wrong in the illusion.

Wow, that is good! I had to work it out mathematically before i saw what was going on. Once i realised that the shapes could not possibly fit together properly in the second arrangement i saw that the angles were indeed slightly distorted.

The true area is 64 squares.

the hypotenuses on the 5×13 are not a straight line. the extra area makes up the 1 square unit difference

Bob and Paulo are correct. If you assume the shapes have been drawn correctly and do the math, the 5 x 13 one actually works out to 64, not 65. The lines between the triangles and the trapezoids have been distorted.

Triangles: 1/2 x base x height = 1/2 x 3 x 8 = 12 each

Trapezoids: 1/2 x (a + b) x height = 1/2 x (3 + 5) x 5 = 20 each.

12 x 2 + 20 x 2 = 24 + 40 =

64paulo segio helped me to spot it but i’m still not sure how it makes that much of a difference. i see though that one of the triangles isnt actually a triangle.

interesting one.

This is the way it is because it is.

I don’t get it.

Its not the overall size that is important – its the area of each shape. There are 2 shapes that have 24 squares, and 2 shapes with 20 squares. 24+24+20+20= 88 squares in both layouts

i could prolly expane this on the spot right now, but nun o yal would listen.

For the pieces to fit perfectly, the slopes of the shapes have to be exact. Calculating the slope of the quadrangle we get 5/2 or 2.5. The slope of the triangle is 7/3 or 2.333. If you used actual pieces put together, there sould be a slight gap which would account for the exta unit of area.

Reducing the size or quality of the illusion doesn’t change the fact that it’s geometrically impossible to fit those shapes together to form the perfect right angles in the 5×13 rectangle. Any freshman math student could sit down and mathematically prove that shape is impossible with true triangles.

It’s the way it’s arranged.. the numbers give it away

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:))

Yeah, like someone has said before, the hypotenuses of the two triangles do not line up with each other. If you do a bunch of calculation, you’ll get the total angle (from those two angles involving in the 2 triangles) to be 190.something degree => that’s the mistake.

If you compare the triangles in the smaller figure with the triangles in the larger figure, you will see that they are not the same and the larger figure actually contains distorted triangles. I can see this from a glance but if you want to prove it, simply count off the rectangle inside the triange formed by 2×3 squares or 6 squares. This rectangle fits inside the triangles in the larger figure but does not fit inside the triangles in the smaller figure. Thus the distorted triangles have each been enlarged enough to fill up an extra .5 of a square. .5 x 2 = 1 which accounts for the extra 1 unit squared.

I like this one.

the line in the middle of 8×13 isnt straight!

There is a lot of talk about distorted this and abnormal that, but I’d like to attempt more of a “math” answer as to why this is impossible.

Assume that each right triangle has a base of 8 and a height of 3 (as in the 8×8 image). This means that the slope on the triangle is 3/8. Or simply that for every 3 the slop moves in either an X or Y direction, that it must also move by 8 in the Y or X direction.

Then consider that the trapezoid has sides 5, 5, and 3. This means the trapezoid has a slope of 2/5.

For the two slops to line up, and all right angles to remain square with the X,Y grid, then the slope of the triangle (3/8) would have to equal the slope of the trapezoid (2/5). And since 3/8 = 15/40 and 2/5 = 16/40, then clearly 16 does not equal 15 so the slopes cannot be equal, and thus we have a (weak) proof that the given rearrangement of the shapes is not possible.

Hence, we have an illusion, which is what this is site is all about anyway. Thanks for the cool mental exercise.

the first geometry illusion that was posted on this site was given to us on a brain teaser page in my AP Physics class. it asked how it was possible.

i said because it wants to.

it makes sense now that its been explained, but its still rather hard to believe.

the triangles in the second box r not triangles. they r slightly curved outward and inward slightly. if the xtra spaces r added up, they equal 1 square.

In the 5×13 image, the 2 shaded shapes (that appear to be right angled triangles until closely examined) are actually quadrilaterals (four sided shapes), the line that appears to be the hypotenuse is actually 2 lines, one being at a slightly different angle.

they are both not rectangles. one is a square and the other is a true rectangle. squares have a larger surface area in general. so the mistake is in thinking they are both rectangles in my perspective.

bob is right. if you look at the line connecting the triangles in the 65 one, the line bends slightly. try putting a ruler or a flat edge up to the line, you can see that it curves.

Hey piece of cake…

They aren’t triangles in the second rectangle (5×13)they are quads.

the line that runs across the one that adds up to 65 is not strait and if the gaps that would be there if it was strait add up to 1

I don’t think the previous explainations are clear, so I will rephrase.

As you can see in the first diagram, the gray shape angles are 2×5. The yellow shape have an angle of 3×8.

These two angles are not the same and the won’t fit each other. In the second picture it shows the two angles together. In reality it wont work, in the second picture there will be a sliver of space between gray and orange shapes. That sliver is equal to the extra space.

Um… I can see where the problem is, but it’s kind of hard to explain…

The shapes of the different shapes (and yes, I do realize that is kind of awkward wording)INSIDE of the bigger shapes are distorted. Anyways, FYI, paulo sergio, distorced is not a word.

To calculate the angle, use arctan 8/3 = 69,4 degrees,

While arctan 5/2 = 68,2 degrees.

So the angles are not the same.

You can even see the the line is not a straight line.

Picture 64 is ok.

Picture 65 has a mistake:

The top line of the left tirangle is 3 units,

and the remaining left seems to be 2.

But the left triangle at this position ist only 1,875.

So there is a (square) gap between the triangles (and a triangle gap between grey and orange).

You can’t see the gap in this picture, but you can calculate it. It’s exactly 1 unit².

I think Bob is right!

the 2 rectangles 5×13 and 8×8 don’t contain the same shapes, but in the 5×13 rectangle there are 2 quadrangles that are very similar to the 2 trinagles in the 8×8 rectangle. So we can’t compare different shapes and try to find the same area.

I did make one of myself.

If you make first the one on the left en move the 4 objects like the one on the right i get a big gap betwen the diagaonal line.

Yea if you look closely at the 65 one the line that the two ‘tiangles’ make is wobbly. Also this illusion is similar to another one i saw on this site which explains in detail what is going on

BAH, Too boring for the morning. I need something more.

The problem on the 65 one is where the two orange triangles are touching, if you look hard enough you can see a kink when they meet, the diagonal goes through 1×3 squares whereas when its an orange and grey shape next to each other the diagonal only goes through 1×2.5 squares.

The shapes in the first 8 by 8 picture do not actually fit together in the way depicted in the 13 by 5 one. The slope of the diagonal side of each of the triangles (yellow) is 8 over 3. 8/3=2 2/3 or 2.6666667. The slope of each of the diagonal sides of the quadrilaterals (gray) is 5 over 2. 5/2=2 1/2 or 2.5. 2.5 does not equal 2.6666667, therefore the shapes cannot be put together contiguously in a way to create a 13 by 5 shape. The reason for the larger area is precisely this disparity of angle, as there should actually be a sliver of empty space from the bottom left to the top right.

In the second figure. If you add the angles of the up right are 88.4546º It is not a right angle, then the main diagonal is not straiht

Bob and Paulo Sergio are right on the ball. Using a straight edge you can see the bend in the hypotenuse of the shapes thus it is no longer a triangle

I still don’t see it. Please explain!

I remember this one from geometry. Bob is close and it is in the angles. The easiest way to see how this works is by getting graph paper and cutting the shapes out. Then put them together in the second configuration. What happens is the angle of the diagonals between the triangle and the quadrelateral don’t line up perfectly because they are not the same. This leaves a thin quadrelateral gap along the diagonal of the 5×13 diagram. The area of this gap is exactly the same as the area of one block and that is what accounts for the increase in area. With line thickness in this drawing they can cover up the gap but if you actually make the figures it’s obviously there.

second design is wrong. The long trasversal line is not a line but is a fragmented line.

Little errors make the square.

Bye

Dave

Nice one

The posed question was, “if the shapes are the same size and area,” but they are not. IN particular the orange triangles in the 64 sq configuration are right triangles with sides 3×8 note the “pitch” or slope is 8 horizontal squares per 3 vertical… or 2 2/3 horizontal for each vertical. In the second picture we can clearly see the pitch change from 3 to 1 in the region where the orange “triangles” touch (the width changes from 3 to 2 in the change of elevation of three) to 5 to 2 or 2.5 to 1 for the remainder of the polygon. This indicates that these are not really triangles but quadralaterals. Since they are not triangles at all, they cannot be the same triangles as in the other picture.

The flaw is in the poor picture quality with border lines too thick to see. If you were to cut shapes in a paper sheet and align them accordingly you would see that in the 5X13 shape there would be a gap area with no paper.

This is the missing area (65-64). But, the gap is so thin that you cant see it in the picture. A simple math test is to divide 5/2 and 8/3. They would be equal in the absence of gap. In this case we get 2 1/2 and 2 2/3 respectively. Because of the different angle you get this area gap. Amusing !

indeed the two figures contain the same 4 components but arrange differently with the 8×8 packed in most optimal way.

furthermore, the difference between the 8×8 cube and the 5×13 rectangular is that the later contains 2 shared angle points each 0.5 (at H5-6:W2-3 and H8-9:W3-4). the addition of those 2 points create the additional square (1×1) and therefore 64+1=65

This one is a bit weak. It is obvious to the eye that both triangles on the right do not build a straight line. In fact, those are not triangles, they are “quadrangles”.

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

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.

Hey, Black Mamba, thank you very much. I have a poor English and I was very tired to look for a dictionary. ;)

I think there is hole in the middle – very long and very narrow.

Ok. Here’s where I found the problem.

The top of the quadrangle equals exactly 3 squares. At the cross section of the triangle where the top of the quadrangle meets. it should equal 2. (2+3=5). If you look at the 8×8 square, The cross section of the triangle at that point is clearly less than 2 squares. I think that’s a sound explanation.

If you actually try to cut up an 8×8 square that way and rearrange the pieces, there’s a really thin hole in the middle of the rectangle which has an area of 1.

I think that explains it.

This one’s just stupid. It’s not an illusion at all, just very sloppy drawing.

There’s a thin parallelogram in the core of the right-side shape, which is 0.117 units wide at it’s thickest. The heavy line weights in this picture help to obscure its presence. They’ve also fudged the shape of the yellow triangles on the right to further obscure it.

Essentially, the ‘diagonal’ lines in the left-side shape are not perpendicular to each other. Instead of 90°, they are 88.7546°. This means they are not collinear in the right-side shape, as you would assume. This creates a 1 square unit parallelogram.

dude,weird post

the shapes are the same, just placed to take up more room!

The angle on the yellow pieces is 20.56 degrees [arctan(3/8)], the angel across the middle of the grey area, not the right-angle corners, is 21.80 degrees [arctan(2/5)].

The 1.25 degree difference actually creates two long thin obtuse triangles that enclose the extra square.

ugh everyones talkin bout angles and junk the solution is even simpler than that. the two orange triangles have the same length when place together to make a square. in the second shape, the two triangles are places farther apart to make a longer length. dummies!

the trangle in the rectangle has a curved edge

count it every 2 it should hit a corner but on 4th one it is compleatly off

Ha!

I calculated the angles one is about 1.3° bigger than the other.

They should be equal

Oh…

Ok everybody already knows…

erm…

YOU FAIL!!!

And everybody talkin bullshit like ashhole…

Better get proper education

@ashhole:

how dumb are you!?!?

just because the rectangels are placed to more length the area isn’t gettin bigger!

ive pasted it onto “paint” and it works

The gray shapes are identical in both pictures, the yellow shapes are different (triangles in the first picture, quadrangles in the second picture). The gray shapes cross two units in five units in both pictures. The yellow shapes cross 3 units in 8 units in a straight line in the first picture. But in the second picture, the yellow shapes first cross 2 units in 5 units along the gray shapes, then 1 unit in 3 units along each other. The wannabe diagonal line in the second picture is broken at 5 units from the top and 5 units from the bottom.

Print it out and get a ruler. It’ll all make sense then. Like the site says, “Optical Illusion”

haha two years later and i comment

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.

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

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

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!!!

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).

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.

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!

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.

Elaine is also right. A group of us are using different words to express the same idea:))

“Same shape” that means same angles same curves.

So all explanations that describe the components as having different angles or curves are automatically eliminated.

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

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.

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.

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

It’s relatively simple math…there is an optimal area for any object to be in so that you have the least amount of area. That just so happens to be a circle, where everything is distributed evenly from the center. When you have a rectangle, there is a farther distance that some sections have to go to make up the whole shape (like the corners), so it’s going to make a larger area.

The orange shapes are not the same, they’re triangles in the 8×8 but in the 5×13 you can see the hypotenuse is bumped out, which makes the extra area.