Round 9: Tossup 1

Constructing this shape with a compass and straightedge is the simplest application of Carlyle (-5[1])circles (-5[1])to constructing polygons. Any polyhedron made of hexagons and these shapes must have exactly 12 of these shapes as faces. (-5[1])The platonic solid with 12 faces has this shape (-5[1])as its (15[1])faces. (15[2])The lengths (-5[1])of the diagonals (15[1])of the regular form (15[1])of these shapes are equal to the (*) golden ratio (10[1])times the side length. These shapes are the regular polygons (0[1])with the smallest number of sides that cannot (-5[1])tile the plane. The interior angles of this (10[2])shape (10[1])are 108 degrees. (10[1]-5[1])For 10 points, name these polygons that have 5 sides. ■END■ (10[5])

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