Squaring the circle is one of the three great problems of Classical Geometry, along with the trisection of the angle and the duplication of the cube. Since 1800 B.C. mathematicians have worked on the problem of constructing a square equal in area to that of a given circle. Whether or not this is possible depends, of course, on what tools you allow yourself. Plato insisted that the problem be solved with straightedge and compass only. To achieve this requires constructing a length equal to sqrt(pi) times the radius of the circle. Thus when Lindemann proved in 1882 that Pi is transcendental (not the root of any polynomial with rational coefficients) he effectively proved that the construction was impossible with only straightedge and compass.

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Click on point C and drag to change the area of the square. Can you make the area approximately equal to that of the circle? What is the length of a side of the square when the areas are equal? How does this compare to the length of the radius of the circle?

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