Round 9: Tossup 1

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

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