The region A shown at right is constructed as follows. The square shown has edges which are unit length. The arcs in the figure are quarter circles of unit radius. The problem has four parts: a) Bound the area of the region from strictly above without using calculus, i.e., use geometry to compute a real number M such that Area(A) < M b) Bound the area of the region from strictly below without using calculus, i.e., use geometry to compute a real number m such that m < Area(A). c) Determine the area of the region A exactly, without using calculus. d) Determine the area of the region A exactly, using calculus.
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.) For more info, please check out:
(Note: None of these links are affiliated with Calculus Humor) 
PollOriginsThis started as a way to express the admins' love of calculus and math in general. As result, this has turned into a gathering place for mathbased humor and weekly challenges. This work by Calculus Humor is licensed under a Creative Commons AttributionNonCommercialShareAlike 3.0 Unported License. Archives
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