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Pythagoras, semicircles and lunes - Lesson+Exploration

A lune is a bidimensional surface bounded by two arcs of a circle having different radii. Hippocrates of Chios, the founder of the Athenian school of geometry, who lived in Athens in 450-420 a.C, was the first mathematician to study the problem of squaring the lunes, and he showed that the ratio of the areas of two circles is equal to the ratio of the squares constructed on their diameters. This result is named squaring because it shows that a curvilinear area is equivalent to another one, bounded by segments, so it can be easily calculated and built with compass and straightedge. The GeoGebra applet shows a result derived from the theorem stated above.