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'Squaring' the Circle

Drag the GOLD dot to 'square' the circle geometrically - when you do this two sets of parallel segments that are perpendicular to one another are connected at the corners by quarter circles. Drag the slider to 'square' the circle algebraically - when you do this a generalization of the circle relation is plotted. What conjectures do you have in each case about ratio of the area of the 'squared circle' to that of the limiting square? What questions could / would you ask your students based on this applet?