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In this paper, constructions of regular pentagon and decagon, and thecalculation of the main trigonometric ratios of the corresponding centralangles are approached. In this way, for didactic purposes, it is intended toshow the reader that it is possible to broaden the study of Trigonometry byaddressing new applications and exercises, for example, with angles 18{\deg},36{\deg} and 72{\deg}. It is also considered constructions of other regularpolygons and a relation to a construction of the regular icosahedron.

Geometric Constructions of Regular Polygons and Applications to Trigonometry