# Where does the gap come from? [duplicate]

Can anyone tell me please where does the gap come from?

Thanks and sorry if the question is not exactly relevant, I just didn't know where else to ask.

## marked as duplicate by Henry, S.C.B., user91500, gebruiker, MacavityApr 28 '16 at 10:50

• The slope angle of the red and blue pieces is different. – almagest Apr 26 '16 at 19:12
• Or, watch carefully the points $(5,2)$ and $(8,3)$ on both pictures. – Berci Apr 26 '16 at 19:27
• Martin Gardner has an informative chapter on this in his book Mathematics, Magic, and Mystery. – Brian Tung Apr 27 '16 at 4:26
• This question gave me a huge craving for infinite chocolate! – heltonbiker Apr 27 '16 at 18:38
• Some people would say this is an optical illusion. Your brain seeing what it expects to see. I say take a lesson from wood shop, pick up the paper, and sight down the lines. You'll see they aren't really straight without doing any math. – candied_orange Apr 27 '16 at 20:35

Figure out the area of the little gap in the middle:

(you can probably guess it, but it's easy to calculate).

Try it for yourself before peeking at the explanation at the bottom.

From there it's easy to see where the area for the square comes from.

Total area of the 13 × 5 region = 65 squares.

Total area of the two unmarked rectangular parts = (5 × 3) × 2 = 30

Total area of blue triangles = (5 × 2 × ½) × 2 = 10

Total area of red triangles = (8 × 3 × ½) × 2 = 24

Remaining area = 65 - 30 - 10 - 24 = 1 square.

If it's still not clear, rotate the upper two triangles each about the center of its own diagonal (which leaves the upper edge of the gap unchanged), you get the triangles from your two diagrams overlaid on each other in the correct positions:

It takes the extra square to make up for the area of that skinny parallelogram-shaped gap between the regions covered by the two arrangements of shapes.

If I did this

Would you ask where the hole came from?