This is a demonstration of a paradoxical dissection that was discovered by an amateur magician named Paul Curry. It is called the missing square puzzle. A right triangle is dissected into four pieces and rearranged, and the new shape appears to be identical to the old shape, except that a square is missing. To see this dissection in action, drag the slider or click the play button.



This trick has a simple explanation. The four pieces do not actually form a right triangle; the hypotenuse is bent inward. After rearranging the pieces, the hypotenuse bends outward. The difference in the bends accounts for the missing unit of area. Click the first checkbox to see that the original triangle is not quite a triangle, or click on the second checkbox to reveal the missing square. (The area of the gray parallelogram keeps the same area as it moves, although the shape changes.)

Another version of this puzzle is here: http://www.geogebratube.org/student/m840
See http://mathblag.wordpress.com/2011/08/28/a-paradoxical-dissection/ for discussion of a similar paradoxical dissection.
Alexander Bogomolny also discusses this paradox. http://www.cut-the-knot.org/Curriculum/Fallacies/CurryParadox.shtml