In various books and internet websites you can find a puzzle called Impossible area puzzle. This puzzle consists of 4 pieces created by cutting one square, and the task is to make rectangle of these four pieces, which is not particularly difficult. Various texts that come with this puzzle say that the surface area when the pieces are arranged into square is not equal with the surface area when the pieces are arranged into rectangle.
Anyone who understands basic mathematic knows that this is impossible, but when we calculate surfaces from our puzzle example on the picture, we get:
Square surface area:
P[mm] = 203,2mm * 203,2mm = 41290,24mm2
P[in] = 8in *8in = 64in2
Rectangle surface area:
P[mm] = 330,2mm * 127mm = 41935,4mm2
P[in] = 13in *5in = 65in2
We can see from the calculation that this is true, but if we calculate the angles of inclined lines, we will see that they differ a little (angles 68,2o and 69,4o - picture above), so there is a gap, which we painted with green color. In books, this puzzle is always drawn without that gap, and if you do not test it mathematically, you are assured that everything is alright. If you make this puzzle of wood, you would think that the gap is caused by inaccurate cutting.
The conclusion is that this puzzle is, of course, mathematically impossible. If you make an effort when making this puzzle, you can almost mask the gap between pieces and then amuse and amaze your friends that are interested in mathematic. If the gap will not be obviously visible, you will have fun, while they will try to understand, how can be possible what they are seeing!
There are a great number of similar puzzles, invented on a same principle, but it is enough to do accurately geometric calculations to realize that they are more optical illusions than refutation of the basic mathematical principles.